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Theorem List for Metamath Proof Explorer - 28801-28900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremsstrALT2 28801 Virtual deduction proof of sstr 3348, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 28800 using the command file translatewithout_overwriting.cmd . It was not minimized because the automated minimization excluding duplicates generates a minimized proof which, although not directly containing any duplicates, indirectly contains a duplicate. That is, the trace back of the minimized proof contains a duplicate. This is undesirable because some step(s) of the minimized proof use the proven theorem. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )
 
TheoremsuctrALT2VD 28802 Virtual deduction proof of suctrALT2 28803. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  Tr  suc  A )
 
TheoremsuctrALT2 28803 Virtual deduction proof of suctr 4656. The sucessor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 28802 using the tools command file translatewithout_overwritingminimize_excludingduplicates.cmd . (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  Tr  suc  A )
 
Theoremelex2VD 28804* Virtual deduction proof of elex2 2960. (Contributed by Alan Sare, 25-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  E. x  x  e.  B )
 
Theoremelex22VD 28805* Virtual deduction proof of elex22 2959. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  B  /\  A  e.  C ) 
 ->  E. x ( x  e.  B  /\  x  e.  C ) )
 
Theoremeqsbc3rVD 28806* Virtual deduction proof of eqsbc3r 3210. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  (
 [. A  /  x ]. C  =  x  <->  C  =  A ) )
 
Theoremzfregs2VD 28807* Virtual deduction proof of zfregs2 7658. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  =/=  (/)  ->  -.  A. x  e.  A  E. y ( y  e.  A  /\  y  e.  x )
 )
 
Theoremtpid3gVD 28808 Virtual deduction proof of tpid3g 3911. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  A  e.  { C ,  D ,  A }
 )
 
Theoremen3lplem1VD 28809* Virtual deduction proof of en3lplem1 7659. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  A  ->  E. y ( y  e.  { A ,  B ,  C }  /\  y  e.  x ) ) )
 
Theoremen3lplem2VD 28810* Virtual deduction proof of en3lplem2 7660. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  E. y
 ( y  e.  { A ,  B ,  C }  /\  y  e.  x ) ) )
 
Theoremen3lpVD 28811 Virtual deduction proof of en3lp 7661. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A )
 
19.24.7  Theorems proved using virtual deduction with mmj2 assistance
 
Theoremsimplbi2VD 28812 Virtual deduction proof of simplbi2 609. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
h1::  |-  ( ph  <->  ( ps  /\  ch ) )
3:1,?: e0_ 28738  |-  ( ( ps  /\  ch )  ->  ph )
qed:3,?: e0_ 28738  |-  ( ps  ->  ( ch  ->  ph ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( ps  /\  ch ) )   =>    |-  ( ps  ->  ( ch  ->  ph ) )
 
Theorem3ornot23VD 28813 Virtual deduction proof of 3ornot23 28446. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::
 |-  (. ( -.  ph  /\  -.  ps )  ->.  ( -.  ph  /\  -.  ps ) ).
2::  |-  (. ( -.  ph  /\  -.  ps ) ,. ( ch  \/  ph  \/  ps )  ->.  ( ch  \/  ph  \/  ps ) ).
3:1,?: e1_ 28582  |-  (. ( -.  ph  /\  -.  ps )  ->.  -.  ph ).
4:1,?: e1_ 28582  |-  (. ( -.  ph  /\  -.  ps )  ->.  -.  ps ).
5:3,4,?: e11 28643  |-  (. ( -.  ph  /\  -.  ps )  ->.  -.  ( ph  \/  ps ) ).
6:2,?: e2 28586  |-  (. ( -.  ph  /\  -.  ps ) ,. ( ch  \/  ph  \/  ps )  ->.  ( ch  \/  ( ph  \/  ps ) ) ).
7:5,6,?: e12 28690  |-  (. ( -.  ph  /\  -.  ps ) ,. ( ch  \/  ph  \/  ps )  ->.  ch ).
8:7:  |-  (. ( -.  ph  /\  -.  ps )  ->.  ( ( ch  \/  ph  \/  ps )  ->  ch ) ).
qed:8:  |-  ( ( -.  ph  /\  -.  ps )  ->  ( ( ch  \/  ph  \/  ps )  ->  ch ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( -.  ph  /\  -.  ps )  ->  ( ( ch  \/  ph  \/  ps )  ->  ch ) )
 
Theoremorbi1rVD 28814 Virtual deduction proof of orbi1r 28447. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ph  <->  ps )  ->.  ( ph  <->  ps ) ).
2::  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ph )  ->.  ( ch  \/  ph ) ).
3:2,?: e2 28586  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ph )  ->.  ( ph  \/  ch ) ).
4:1,3,?: e12 28690  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ph )  ->.  ( ps  \/  ch ) ).
5:4,?: e2 28586  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ph )  ->.  ( ch  \/  ps ) ).
6:5:  |-  (. ( ph  <->  ps )  ->.  ( ( ch  \/  ph )  ->  ( ch  \/  ps ) ) ).
7::  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ch  \/  ps ) ).
8:7,?: e2 28586  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ps  \/  ch ) ).
9:1,8,?: e12 28690  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ph  \/  ch ) ).
10:9,?: e2 28586  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ch  \/  ph ) ).
11:10:  |-  (. ( ph  <->  ps )  ->.  ( ( ch  \/  ps )  ->  ( ch  \/  ph ) ) ).
12:6,11,?: e11 28643  |-  (. ( ph  <->  ps )  ->.  ( ( ch  \/  ph )  <->  ( ch  \/  ps ) ) ).
qed:12:  |-  ( ( ph  <->  ps )  ->  ( ( ch  \/  ph )  <->  ( ch  \/  ps ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ch  \/  ph ) 
 <->  ( ch  \/  ps ) ) )
 
Theorembitr3VD 28815 Virtual deduction proof of bitr3 28448. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ph  <->  ps )  ->.  ( ph  <->  ps ) ).
2:1,?: e1_ 28582  |-  (. ( ph  <->  ps )  ->.  ( ps  <->  ph ) ).
3::  |-  (. ( ph  <->  ps ) ,. ( ph  <->  ch )  ->.  ( ph  <->  ch ) ).
4:3,?: e2 28586  |-  (. ( ph  <->  ps ) ,. ( ph  <->  ch )  ->.  ( ch  <->  ph ) ).
5:2,4,?: e12 28690  |-  (. ( ph  <->  ps ) ,. ( ph  <->  ch )  ->.  ( ps  <->  ch ) ).
6:5:  |-  (. ( ph  <->  ps )  ->.  ( ( ph  <->  ch )  ->  ( ps  <->  ch ) ) ).
qed:6:  |-  ( ( ph  <->  ps )  ->  ( ( ph  <->  ch )  ->  ( ps  <->  ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ph  <->  ch )  ->  ( ps 
 <->  ch ) ) )
 
Theorem3orbi123VD 28816 Virtual deduction proof of 3orbi123 28449. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) ) ).
2:1,?: e1_ 28582  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ph  <->  ps ) ).
3:1,?: e1_ 28582  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ch  <->  th ) ).
4:1,?: e1_ 28582  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ta  <->  et ) ).
5:2,3,?: e11 28643  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ph  \/  ch )  <->  ( ps  \/  th ) ) ).
6:5,4,?: e11 28643  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ( ph  \/  ch )  \/  ta )  <->  ( ( ps  \/  th )  \/  et ) ) ).
7:?:  |-  ( ( ( ph  \/  ch )  \/  ta )  <->  ( ph  \/  ch  \/  ta ) )
8:6,7,?: e10 28649  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ph  \/  ch  \/  ta )  <->  ( ( ps  \/  th )  \/  et ) ) ).
9:?:  |-  ( ( ( ps  \/  th )  \/  et )  <->  ( ps  \/  th  \/  et ) )
10:8,9,?: e10 28649  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ph  \/  ch  \/  ta )  <->  ( ps  \/  th  \/  et ) ) ).
qed:10:  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->  ( ( ph  \/  ch  \/  ta )  <->  ( ps  \/  th  \/  et ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th )  /\  ( ta 
 <->  et ) )  ->  ( ( ph  \/  ch 
 \/  ta )  <->  ( ps  \/  th 
 \/  et ) ) )
 
Theoremsbc3orgVD 28817 Virtual deduction proof of sbc3org 28471. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2:1,?: e1_ 28582  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ( ph  \/  ps )  \/  ch )  <->  ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch ) ) ).
3::  |-  ( ( ( ph  \/  ps )  \/  ch )  <->  ( ph  \/  ps  \/  ch ) )
32:3:  |-  A. x ( ( ( ph  \/  ps )  \/  ch )  <->  ( ph  \/  ps  \/  ch ) )
33:1,32,?: e10 28649  |-  (. A  e.  B  ->.  [. A  /  x ]. ( ( ( ph  \/  ps )  \/  ch )  <->  ( ph  \/  ps  \/  ch ) ) ).
4:1,33,?: e11 28643  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ( ph  \/  ps )  \/  ch )  <->  [. A  /  x ]. ( ph  \/  ps  \/  ch ) ) ).
5:2,4,?: e11 28643  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch ) ) ).
6:1,?: e1_ 28582  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) ).
7:6,?: e1_ 28582  |-  (. A  e.  B  ->.  ( ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) ).
8:5,7,?: e11 28643  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) ).
9:?:  |-  ( ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) )
10:8,9,?: e10 28649  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) ) ).
qed:10:  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  (
 [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ].
 ch ) ) )
 
Theorem19.21a3con13vVD 28818* Virtual deduction proof of alrim3con13v 28472. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ph  ->  A. x ph )  ->.  ( ph  ->  A. x ph ) ).
2::  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ( ps  /\  ph  /\  ch ) ).
3:2,?: e2 28586  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ps ).
4:2,?: e2 28586  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ph ).
5:2,?: e2 28586  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ch ).
6:1,4,?: e12 28690  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ph ).
7:3,?: e2 28586  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ps ).
8:5,?: e2 28586  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ch ).
9:7,6,8,?: e222 28591  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ( A. x ps  /\  A. x ph  /\  A. x ch ) ).
10:9,?: e2 28586  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ( ps  /\  ph  /\  ch ) ).
11:10:in2  |-  (. ( ph  ->  A. x ph )  ->.  ( ( ps  /\  ph  /\  ch )  ->  A. x ( ps  /\  ph  /\  ch ) ) ).
qed:11:in1  |-  ( ( ph  ->  A. x ph )  ->  ( ( ps  /\  ph  /\  ch )  ->  A. x ( ps  /\  ph  /\  ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  A. x ph )  ->  ( ( ps  /\  ph  /\  ch )  ->  A. x ( ps 
 /\  ph  /\  ch )
 ) )
 
TheoremexbirVD 28819 Virtual deduction proof of exbir 1374. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) )  ->.  ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ).
2::  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ,.  ( ph  /\  ps )  ->.  ( ph  /\  ps ) ).
3::  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ,.  ( ph  /\  ps ) ,  th  ->.  th ).
5:1,2,?: e12 28690  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ,  ( ph  /\  ps )  ->.  ( ch  <->  th ) ).
6:3,5,?: e32 28724  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ,  ( ph  /\  ps ) ,  th  ->.  ch ).
7:6:  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ,  ( ph  /\  ps )  ->.  ( th  ->  ch ) ).
8:7:  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) )  ->.  ( ( ph  /\  ps )  ->  ( th  ->  ch ) ) ).
9:8,?: e1_ 28582  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) )  ->.  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) ) ).
qed:9:  |-  ( ( ( ph  /\  ps )  ->  ( ch  <->  th ) )  ->  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) ) )
(Contributed by Alan Sare, 13-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  /\  ps )  ->  ( ch  <->  th ) )  ->  ( ph  ->  ( ps  ->  ( th  ->  ch )
 ) ) )
 
TheoremexbiriVD 28820 Virtual deduction proof of exbiri 606. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
h1::  |-  ( ( ph  /\  ps )  ->  ( ch  <->  th ) )
2::  |-  (. ph  ->.  ph ).
3::  |-  (. ph ,. ps  ->.  ps ).
4::  |-  (. ph ,. ps ,. th  ->.  th ).
5:2,1,?: e10 28649  |-  (. ph  ->.  ( ps  ->  ( ch  <->  th ) ) ).
6:3,5,?: e21 28696  |-  (. ph ,. ps  ->.  ( ch  <->  th ) ).
7:4,6,?: e32 28724  |-  (. ph ,. ps ,. th  ->.  ch ).
8:7:  |-  (. ph ,. ps  ->.  ( th  ->  ch ) ).
9:8:  |-  (. ph  ->.  ( ps  ->  ( th  ->  ch ) ) ).
qed:9:  |-  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps )  ->  ( ch  <->  th ) )   =>    |-  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) )
 
Theoremrspsbc2VD 28821* Virtual deduction proof of rspsbc2 28473. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2::  |-  (. A  e.  B ,. C  e.  D  ->.  C  e.  D ).
3::  |-  (. A  e.  B ,. C  e.  D ,. A. x  e.  B  A. y  e.  D ph  ->.  A. x  e.  B A. y  e.  D ph ).
4:1,3,?: e13 28714  |-  (. A  e.  B ,. C  e.  D ,. A. x  e.  B  A. y  e.  D ph  ->.  [. A  /  x ]. A. y  e.  D ph ).
5:1,4,?: e13 28714  |-  (. A  e.  B ,. C  e.  D ,. A. x  e.  B  A. y  e.  D ph  ->.  A. y  e.  D [. A  /  x ]. ph ).
6:2,5,?: e23 28721  |-  (. A  e.  B ,. C  e.  D ,. A. x  e.  B  A. y  e.  D ph  ->.  [. C  /  y ]. [. A  /  x ]. ph ).
7:6:  |-  (. A  e.  B ,. C  e.  D  ->.  ( A. x  e.  B  A. y  e.  D ph  ->  [. C  /  y ]. [. A  /  x ]. ph ) ).
8:7:  |-  (. A  e.  B  ->.  ( C  e.  D  ->  ( A. x  e.  B A. y  e.  D ph  ->  [. C  /  y ]. [. A  /  x ]. ph ) ) ).
qed:8:  |-  ( A  e.  B  ->  ( C  e.  D  ->  ( A. x  e.  B A. y  e.  D ph  ->  [. C  /  y ]. [. A  /  x ]. ph ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  ( C  e.  D  ->  (
 A. x  e.  B  A. y  e.  D  ph  -> 
 [. C  /  y ]. [. A  /  x ].
 ph ) ) )
 
Theorem3impexpVD 28822 Virtual deduction proof of 3impexp 1375. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ph  /\  ps  /\  ch )  ->  th ) ).
2::  |-  ( ( ph  /\  ps  /\  ch )  <->  ( ( ph  /\  ps )  /\  ch ) )
3:1,2,?: e10 28649  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ( ph  /\  ps )  /\  ch )  ->  th ) ).
4:3,?: e1_ 28582  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ph  /\  ps )  ->  ( ch  ->  th ) ) ).
5:4,?: e1_ 28582  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) ).
6:5:  |-  ( ( ( ph  /\  ps  /\  ch )  ->  th )  ->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
7::  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) ).
8:7,?: e1_ 28582  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ph  /\  ps )  ->  ( ch  ->  th ) ) ).
9:8,?: e1_ 28582  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ( ph  /\  ps )  /\  ch )  ->  th ) ).
10:2,9,?: e01 28646  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  th ) ).
11:10:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->  ( ( ph  /\  ps  /\  ch )  ->  th ) )
qed:6,11,?: e00 28734  |-  ( ( ( ph  /\  ps  /\  ch )  ->  th )  <->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  /\  ps  /\ 
 ch )  ->  th )  <->  (
 ph  ->  ( ps  ->  ( ch  ->  th )
 ) ) )
 
Theorem3impexpbicomVD 28823 Virtual deduction proof of 3impexpbicom 1376. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) ) ).
2::  |-  ( ( th  <->  ta )  <->  ( ta  <->  th ) )
3:1,2,?: e10 28649  |-  (. ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  ( ta  <->  th ) ) ).
4:3,?: e1_ 28582  |-  (. ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )  ->.  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) ) ).
5:4:  |-  ( ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )  ->  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) ) )
6::  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )  ->.  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) ) ).
7:6,?: e1_ 28582  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  ( ta  <->  th ) ) ).
8:7,2,?: e10 28649  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) ) ).
9:8:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )  ->  ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) ) )
qed:5,9,?: e00 28734  |-  ( ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  /\  ps  /\ 
 ch )  ->  ( th 
 <->  ta ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) ) )
 
Theorem3impexpbicomiVD 28824 Virtual deduction proof of 3impexpbicomi 1377. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
h1::  |-  ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )
qed:1,?: e0_ 28738  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )
 
TheoremsbcoreleleqVD 28825* Virtual deduction proof of sbcoreleleq 28474. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2:1,?: e1_ 28582  |-  (. A  e.  B  ->.  ( [. A  /  y ]. x  e.  y  <->  x  e.  A ) ).
3:1,?: e1_ 28582  |-  (. A  e.  B  ->.  ( [. A  /  y ]. y  e.  x  <->  A  e.  x ) ).
4:1,?: e1_ 28582  |-  (. A  e.  B  ->.  ( [. A  /  y ]. x  =  y  <->  x  =  A ) ).
5:2,3,4,?: e111 28629  |-  (. A  e.  B  ->.  ( ( x  e.  A  \/  A  e.  x  \/  x  =  A )  <->  ( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y ) ) ).
6:1,?: e1_ 28582  |-  (. A  e.  B  ->.  ( [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y )  <->  ( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y ) ) ).
7:5,6: e11 28643  |-  (. A  e.  B  ->.  ( [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y )  <->  ( x  e.  A  \/  A  e.  x  \/  x  =  A ) ) ).
qed:7:  |-  ( A  e.  B  ->  ( [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y )  <->  ( x  e.  A  \/  A  e.  x  \/  x  =  A ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  (
 [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y
 ) 
 <->  ( x  e.  A  \/  A  e.  x  \/  x  =  A )
 ) )
 
Theoremhbra2VD 28826* Virtual deduction proof of nfra2 2752. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  ( A. y  e.  B A. x  e.  A ph  ->  A. y A. y  e.  B A. x  e.  A ph )
2::  |-  ( A. x  e.  A A. y  e.  B ph  <->  A. y  e.  B A. x  e.  A ph )
3:1,2,?: e00 28734  |-  ( A. x  e.  A A. y  e.  B ph  ->  A. y A. y  e.  B A. x  e.  A ph )
4:2:  |-  A. y ( A. x  e.  A A. y  e.  B ph  <->  A. y  e.  B A. x  e.  A ph )
5:4,?: e0_ 28738  |-  ( A. y A. x  e.  A A. y  e.  B ph  <->  A. y A. y  e.  B A. x  e.  A ph )
qed:3,5,?: e00 28734  |-  ( A. x  e.  A A. y  e.  B ph  ->  A. y A. x  e.  A A. y  e.  B ph )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  e.  A  A. y  e.  B  ph  ->  A. y A. x  e.  A  A. y  e.  B  ph )
 
TheoremtratrbVD 28827* Virtual deduction proof of tratrb 28475. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  ( Tr  A  /\  A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ).
2:1,?: e1_ 28582  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  Tr  A ).
3:1,?: e1_ 28582  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) ).
4:1,?: e1_ 28582  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  B  e.  A ).
5::  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  ( x  e.  y  /\  y  e.  B ) ).
6:5,?: e2 28586  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  x  e.  y ).
7:5,?: e2 28586  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  y  e.  B ).
8:2,7,4,?: e121 28611  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  y  e.  A ).
9:2,6,8,?: e122 28608  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  x  e.  A ).
10::  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B ) ,  B  e.  x  ->.  B  e.  x ).
11:6,7,10,?: e223 28590  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B ) ,  B  e.  x  ->.  ( x  e.  y  /\  y  e.  B  /\  B  e.  x ) ).
12:11:  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  ( B  e.  x  ->  ( x  e.  y  /\  y  e.  B  /\  B  e.  x ) ) ).
13::  |-  -.  ( x  e.  y  /\  y  e.  B  /\  B  e.  x )
14:12,13,?: e20 28693  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  -.  B  e.  x ).
15::  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B ) ,  x  =  B  ->.  x  =  B ).
16:7,15,?: e23 28721  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B ) ,  x  =  B  ->.  y  e.  x ).
17:6,16,?: e23 28721  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B ) ,  x  =  B  ->.  ( x  e.  y  /\  y  e.  x ) ).
18:17:  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  ( x  =  B  ->  ( x  e.  y  /\  y  e.  x ) ) ).
19::  |-  -.  ( x  e.  y  /\  y  e.  x )
20:18,19,?: e20 28693  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  -.  x  =  B ).
21:3,?: e1_ 28582  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  A. y  e.  A  A. x  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) ).
22:21,9,4,?: e121 28611  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  [. x  /  x ]. [. B  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y ) ).
23:22,?: e2 28586  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  [. B  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y ) ).
24:4,23,?: e12 28690  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  ( x  e.  B  \/  B  e.  x  \/  x  =  B ) ).
25:14,20,24,?: e222 28591  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  x  e.  B ).
26:25:  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  ( ( x  e.  y  /\  y  e.  B )  ->  x  e.  B ) ).
27::  |-  ( A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  ->  A. y A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
28:27,?: e0_ 28738  |-  ( ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->  A. y ( Tr  A  /\  A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) )
29:28,26:  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  A. y ( ( x  e.  y  /\  y  e.  B )  ->  x  e.  B ) ).
30::  |-  ( A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  ->  A. x A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
31:30,?: e0_ 28738  |-  ( ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->  A. x ( Tr  A  /\  A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) )
32:31,29:  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  A. x  A. y ( ( x  e.  y  /\  y  e.  B )  ->  x  e.  B ) ).
33:32,?: e1_ 28582  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  Tr  B ).
qed:33:  |-  ( ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->  Tr  B )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( Tr  A  /\  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) 
 ->  Tr  B )
 
Theorem3ax5VD 28828 Virtual deduction proof of 3ax5 28476. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. A. x ( ph  ->  ( ps  ->  ch ) )  ->.  A. x ( ph  ->  ( ps  ->  ch ) ) ).
2:1,?: e1_ 28582  |-  (. A. x ( ph  ->  ( ps  ->  ch ) )  ->.  ( A. x ph  ->  A. x ( ps  ->  ch ) ) ).
3::  |-  ( A. x ( ps  ->  ch )  ->  ( A. x ps  ->  A. x ch ) )
4:2,3,?: e10 28649  |-  (. A. x ( ph  ->  ( ps  ->  ch ) )  ->.  ( A. x ph  ->  ( A. x ps  ->  A. x ch ) ) ).
qed:4:  |-  ( A. x ( ph  ->  ( ps  ->  ch ) )  ->  ( A. x ph  ->  ( A. x ps  ->  A. x ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x ( ph  ->  ( ps  ->  ch )
 )  ->  ( A. x ph  ->  ( A. x ps  ->  A. x ch ) ) )
 
Theoremsyl5impVD 28829 Virtual deduction proof of syl5imp 28450. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ph  ->  ( ps  ->  ch ) )  ->.  ( ph  ->  ( ps  ->  ch ) ) ).
2:1,?: e1_ 28582  |-  (. ( ph  ->  ( ps  ->  ch ) )  ->.  ( ps  ->  ( ph  ->  ch ) ) ).
3::  |-  (. ( ph  ->  ( ps  ->  ch ) ) ,. ( th  ->  ps )  ->.  ( th  ->  ps ) ).
4:3,2,?: e21 28696  |-  (. ( ph  ->  ( ps  ->  ch ) ) ,. ( th  ->  ps )  ->.  ( th  ->  ( ph  ->  ch ) ) ).
5:4,?: e2 28586  |-  (. ( ph  ->  ( ps  ->  ch ) ) ,. ( th  ->  ps )  ->.  ( ph  ->  ( th  ->  ch ) ) ).
6:5:  |-  (. ( ph  ->  ( ps  ->  ch ) )  ->.  ( ( th  ->  ps )  ->  ( ph  ->  ( th  ->  ch ) ) ) ).
qed:6:  |-  ( ( ph  ->  ( ps  ->  ch ) )  ->  ( ( th  ->  ps )  ->  ( ph  ->  ( th  ->  ch ) ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps 
 ->  ch ) )  ->  ( ( th  ->  ps )  ->  ( ph  ->  ( th  ->  ch )
 ) ) )
 
TheoremidiVD 28830 Virtual deduction proof of idi 2. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
h1::  |-  ph
qed:1,?: e0_ 28738  |-  ph
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   =>    |-  ph
 
TheoremancomsimpVD 28831 Closed form of ancoms 440. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  ( ( ph  /\  ps )  <->  ( ps  /\  ph ) )
qed:1,?: e0_ 28738  |-  ( ( ( ph  /\  ps )  ->  ch )  <->  ( ( ps  /\  ph )  ->  ch ) )
(Contributed by Alan Sare, 25-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  /\  ps )  ->  ch )  <->  ( ( ps 
 /\  ph )  ->  ch )
 )
 
Theoremssralv2VD 28832* Quantification restricted to a subclass for two quantifiers. ssralv 3399 for two quantifiers. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. ssralv2 28470 is ssralv2VD 28832 without virtual deductions and was automatically derived from ssralv2VD 28832.
1::  |-  (. ( A  C_  B  /\  C  C_  D )  ->.  ( A  C_  B  /\  C  C_  D ) ).
2::  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph  ->.  A. x  e.  B A. y  e.  D ph ).
3:1:  |-  (. ( A  C_  B  /\  C  C_  D )  ->.  A  C_  B ).
4:3,2:  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph  ->.  A. x  e.  A A. y  e.  D ph ).
5:4:  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph  ->.  A. x ( x  e.  A  ->  A. y  e.  D ph ) ).
6:5:  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph  ->.  ( x  e.  A  ->  A. y  e.  D ph ) ).
7::  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph ,  x  e.  A  ->.  x  e.  A ).
8:7,6:  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph ,  x  e.  A  ->.  A. y  e.  D ph ).
9:1:  |-  (. ( A  C_  B  /\  C  C_  D )  ->.  C  C_  D ).
10:9,8:  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph ,  x  e.  A  ->.  A. y  e.  C ph ).
11:10:  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph  ->.  ( x  e.  A  ->  A. y  e.  C ph ) ).
12::  |-  ( ( A  C_  B  /\  C  C_  D )  ->  A. x ( A  C_  B  /\  C  C_  D ) )
13::  |-  ( A. x  e.  B A. y  e.  D ph  ->  A. x A. x  e.  B A. y  e.  D ph )
14:12,13,11:  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph  ->.  A. x ( x  e.  A  ->  A. y  e.  C ph ) ).
15:14:  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph  ->.  A. x  e.  A A. y  e.  C ph ).
16:15:  |-  (. ( A  C_  B  /\  C  C_  D )  ->.  ( A. x  e.  B A. y  e.  D ph  ->  A. x  e.  A A. y  e.  C ph ) ).
qed:16:  |-  ( ( A  C_  B  /\  C  C_  D )  ->  ( A. x  e.  B A. y  e.  D ph  ->  A. x  e.  A A. y  e.  C ph ) )
(Contributed by Alan Sare, 10-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  C_  B  /\  C  C_  D )  ->  ( A. x  e.  B  A. y  e.  D  ph  ->  A. x  e.  A  A. y  e.  C  ph ) )
 
TheoremordelordALTVD 28833 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 4595 using the Axiom of Regularity indirectly through dford2 7564. dford2 is a weaker definition of ordinal number. Given the Axiom of Regularity, it need not be assumed that  _E  Fr  A because this is inferred by the Axiom of Regularity. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. ordelordALT 28477 is ordelordALTVD 28833 without virtual deductions and was automatically derived from ordelordALTVD 28833 using the tools program translate..without..overwriting.cmd and Metamath's minimize command.
1::  |-  (. ( Ord  A  /\  B  e.  A )  ->.  ( Ord  A  /\  B  e.  A ) ).
2:1:  |-  (. ( Ord  A  /\  B  e.  A )  ->.  Ord  A ).
3:1:  |-  (. ( Ord  A  /\  B  e.  A )  ->.  B  e.  A ).
4:2:  |-  (. ( Ord  A  /\  B  e.  A )  ->.  Tr  A ).
5:2:  |-  (. ( Ord  A  /\  B  e.  A )  ->.  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  x  =  y  \/  y  e.  x ) ).
6:4,3:  |-  (. ( Ord  A  /\  B  e.  A )  ->.  B  C_  A ).
7:6,6,5:  |-  (. ( Ord  A  /\  B  e.  A )  ->.  A. x  e.  B  A. y  e.  B ( x  e.  y  \/  x  =  y  \/  y  e.  x ) ).
8::  |-  ( ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <->  ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
9:8:  |-  A. y ( ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <->  ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
10:9:  |-  A. y  e.  A ( ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <->  ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
11:10:  |-  ( A. y  e.  A ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <->  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
12:11:  |-  A. x ( A. y  e.  A ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <->  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
13:12:  |-  A. x  e.  A ( A. y  e.  A ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <->  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
14:13:  |-  ( A. x  e.  A A. y  e.  A ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <->  A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
15:14,5:  |-  (. ( Ord  A  /\  B  e.  A )  ->.  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) ).
16:4,15,3:  |-  (. ( Ord  A  /\  B  e.  A )  ->.  Tr  B ).
17:16,7:  |-  (. ( Ord  A  /\  B  e.  A )  ->.  Ord  B ).
qed:17:  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
(Contributed by Alan Sare, 12-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( Ord  A  /\  B  e.  A )  ->  Ord  B )
 
TheoremequncomVD 28834 If a class equals the union of two other classes, then it equals the union of those two classes commuted. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. equncom 3484 is equncomVD 28834 without virtual deductions and was automatically derived from equncomVD 28834.
1::  |-  (. A  =  ( B  u.  C )  ->.  A  =  ( B  u.  C ) ).
2::  |-  ( B  u.  C )  =  ( C  u.  B )
3:1,2:  |-  (. A  =  ( B  u.  C )  ->.  A  =  ( C  u.  B ) ).
4:3:  |-  ( A  =  ( B  u.  C )  ->  A  =  ( C  u.  B ) )
5::  |-  (. A  =  ( C  u.  B )  ->.  A  =  ( C  u.  B ) ).
6:5,2:  |-  (. A  =  ( C  u.  B )  ->.  A  =  ( B  u.  C ) ).
7:6:  |-  ( A  =  ( C  u.  B )  ->  A  =  ( B  u.  C ) )
8:4,7:  |-  ( A  =  ( B  u.  C )  <->  A  =  ( C  u.  B ) )
(Contributed by Alan Sare, 17-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  =  ( B  u.  C )  <->  A  =  ( C  u.  B ) )
 
TheoremequncomiVD 28835 Inference form of equncom 3484. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. equncomi 3485 is equncomiVD 28835 without virtual deductions and was automatically derived from equncomiVD 28835.
h1::  |-  A  =  ( B  u.  C )
qed:1:  |-  A  =  ( C  u.  B )
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  =  ( B  u.  C )   =>    |-  A  =  ( C  u.  B )
 
TheoremsucidALTVD 28836 A set belongs to its successor. Alternate proof of sucid 4652. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sucidALT 28837 is sucidALTVD 28836 without virtual deductions and was automatically derived from sucidALTVD 28836. This proof illustrates that completeusersproof.cmd will generate a Metamath proof from any User's Proof which is "conventional" in the sense that no step is a virtual deduction, provided that all necessary unification theorems and transformation deductions are in set.mm. completeusersproof.cmd automatically converts such a conventional proof into a Virtual Deduction proof for which each step happens to be a 0-virtual hypothesis virtual deduction. The user does not need to search for reference theorem labels or deduction labels nor does he(she) need to use theorems and deductions which unify with reference theorems and deductions in set.mm. All that is necessary is that each theorem or deduction of the User's Proof unifies with some reference theorem or deduction in set.mm or is a semantic variation of some theorem or deduction which unifies with some reference theorem or deduction in set.mm. The definition of "semantic variation" has not been precisely defined. If it is obvious that a theorem or deduction has the same meaning as another theorem or deduction, then it is a semantic variation of the latter theorem or deduction. For example, step 4 of the User's Proof is a semantic variation of the definition (axiom)  suc  A  =  ( A  u.  { A } ), which unifies with df-suc 4579, a reference definition (axiom) in set.mm. Also, a theorem or deduction is said to be a semantic variation of another theorem or deduction if it is obvious upon cursory inspection that it has the same meaning as a weaker form of the latter theorem or deduction. For example, the deduction  Ord  A infers  A. x  e.  A A. y  e.  A ( x  e.  y  \/  x  =  y  \/  y  e.  x ) is a semantic variation of the theorem  ( Ord  A  <->  ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  x  =  y  \/  y  e.  x ) ) ), which unifies with the set.mm reference definition (axiom) dford2 7564.
h1::  |-  A  e.  _V
2:1:  |-  A  e.  { A }
3:2:  |-  A  e.  ( { A }  u.  A )
4::  |-  suc  A  =  ( { A }  u.  A )
qed:3,4:  |-  A  e.  suc  A
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  A  e.  suc  A
 
TheoremsucidALT 28837 A set belongs to its successor. This proof was automatically derived from sucidALTVD 28836 using translatewithout_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  A  e.  suc  A
 
TheoremsucidVD 28838 A set belongs to its successor. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sucid 4652 is sucidVD 28838 without virtual deductions and was automatically derived from sucidVD 28838.
h1::  |-  A  e.  _V
2:1:  |-  A  e.  { A }
3:2:  |-  A  e.  ( A  u.  { A } )
4::  |-  suc  A  =  ( A  u.  { A } )
qed:3,4:  |-  A  e.  suc  A
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  A  e.  suc  A
 
Theoremimbi12VD 28839 Implication form of imbi12i 317. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. imbi12 28458 is imbi12VD 28839 without virtual deductions and was automatically derived from imbi12VD 28839.
1::  |-  (. ( ph  <->  ps )  ->.  ( ph  <->  ps ) ).
2::  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th )  ->.  ( ch  <->  th ) ).
3::  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ph  ->  ch )  ->.  ( ph  ->  ch ) ).
4:1,3:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ph  ->  ch )  ->.  ( ps  ->  ch ) ).
5:2,4:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ph  ->  ch )  ->.  ( ps  ->  th ) ).
6:5:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th )  ->.  ( ( ph  ->  ch )  ->  ( ps  ->  th ) ) ).
7::  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ps  ->  th )  ->.  ( ps  ->  th ) ).
8:1,7:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ps  ->  th )  ->.  ( ph  ->  th ) ).
9:2,8:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ps  ->  th )  ->.  ( ph  ->  ch ) ).
10:9:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th )  ->.  ( ( ps  ->  th )  ->  ( ph  ->  ch ) ) ).
11:6,10:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th )  ->.  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) ).
12:11:  |-  (. ( ph  <->  ps )  ->.  ( ( ch  <->  th )  ->  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) ) ).
qed:12:  |-  ( ( ph  <->  ps )  ->  ( ( ch  <->  th )  ->  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) ) )
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ch  <->  th )  ->  (
 ( ph  ->  ch )  <->  ( ps  ->  th )
 ) ) )
 
Theoremimbi13VD 28840 Join three logical equivalences to form equivalence of implications. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. imbi13 28459 is imbi13VD 28840 without virtual deductions and was automatically derived from imbi13VD 28840.
1::  |-  (. ( ph  <->  ps )  ->.  ( ph  <->  ps ) ).
2::  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th )  ->.  ( ch  <->  th ) ).
3::  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ta  <->  et )  ->.  ( ta  <->  et ) ).
4:2,3:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ta  <->  et )  ->.  ( ( ch  ->  ta )  <->  ( th  ->  et ) ) ).
5:1,4:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ta  <->  et )  ->.  ( ( ph  ->  ( ch  ->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ).
6:5:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th )  ->.  ( ( ta  <->  et )  ->  ( ( ph  ->  ( ch  ->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ) ).
7:6:  |-  (. ( ph  <->  ps )  ->.  ( ( ch  <->  th )  ->  ( ( ta  <->  et )  ->  ( ( ph  ->  ( ch  ->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ) ) ).
qed:7:  |-  ( ( ph  <->  ps )  ->  ( ( ch  <->  th )  ->  ( ( ta  <->  et )  ->  ( ( ph  ->  ( ch  ->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ) ) )
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ch  <->  th )  ->  (
 ( ta  <->  et )  ->  (
 ( ph  ->  ( ch 
 ->  ta ) )  <->  ( ps  ->  ( th  ->  et )
 ) ) ) ) )
 
Theoremsbcim2gVD 28841 Distribution of class substitution over a left-nested implication. Similar to sbcimg 3194. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcim2g 28478 is sbcim2gVD 28841 without virtual deductions and was automatically derived from sbcim2gVD 28841.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2::  |-  (. A  e.  B ,. [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  ->.  [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) ) ).
3:1,2:  |-  (. A  e.  B ,. [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  ->.  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch ) ) ).
4:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ps  ->  ch )  <->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ).
5:3,4:  |-  (. A  e.  B ,. [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  ->.  ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ).
6:5:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  ->  ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ) ).
7::  |-  (. A  e.  B ,. ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )  ->.  ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ).
8:4,7:  |-  (. A  e.  B ,. ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )  ->.  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch ) ) ).
9:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch ) ) ) ).
10:8,9:  |-  (. A  e.  B ,. ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )  ->.  [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) ) ).
11:10:  |-  (. A  e.  B  ->.  ( ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )  ->  [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) ) ) ).
12:6,11:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <->  ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ) ).
qed:12:  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <->  ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ) )
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  (
 [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <->  ( [. A  /  x ]. ph  ->  (
 [. A  /  x ].
 ps  ->  [. A  /  x ].
 ch ) ) ) )
 
TheoremsbcbiVD 28842 Implication form of sbcbiiOLD 3209. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcbi 28479 is sbcbiVD 28842 without virtual deductions and was automatically derived from sbcbiVD 28842.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2::  |-  (. A  e.  B ,. A. x ( ph  <->  ps )  ->.  A. x ( ph  <->  ps ) ).
3:1,2:  |-  (. A  e.  B ,. A. x ( ph  <->  ps )  ->.  [. A  /  x ]. ( ph  <->  ps ) ).
4:1,3:  |-  (. A  e.  B ,. A. x ( ph  <->  ps )  ->.  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ).
5:4:  |-  (. A  e.  B  ->.  ( A. x ( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) ).
qed:5:  |-  ( A  e.  B  ->  ( A. x ( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  (
 A. x ( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ].
 ps ) ) )
 
TheoremtrsbcVD 28843* Formula-building inference rule for class substitution, substituting a class variable for the set variable of the transitivity predicate. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. trsbc 28480 is trsbcVD 28843 without virtual deductions and was automatically derived from trsbcVD 28843.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. z  e.  y  <->  z  e.  y ) ).
3:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. y  e.  x  <->  y  e.  A ) ).
4:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. z  e.  x  <->  z  e.  A ) ).
5:1,2,3,4:  |-  (. A  e.  B  ->.  ( ( [. A  /  x ]. z  e.  y  ->  ( [. A  /  x ]. y  e.  x  ->  [. A  /  x ]. z  e.  x ) )  <->  ( z  e.  y  ->  ( y  e.  A  ->  z  e.  A ) ) ) ).
6:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( z  e.  y  ->  ( y  e.  x  ->  z  e.  x ) )  <->  ( [. A  /  x ]. z  e.  y  ->  ( [. A  /  x ]. y  e.  x  ->  [. A  /  x ]. z  e.  x ) ) ) ).
7:5,6:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( z  e.  y  ->  ( y  e.  x  ->  z  e.  x ) )  <->  ( z  e.  y  ->  ( y  e.  A  ->  z  e.  A ) ) ) ).
8::  |-  ( ( z  e.  y  ->  ( y  e.  A  ->  z  e.  A ) )  <->  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
9:7,8:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( z  e.  y  ->  ( y  e.  x  ->  z  e.  x ) )  <->  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ) ).
10::  |-  ( ( z  e.  y  ->  ( y  e.  x  ->  z  e.  x ) )  <->  ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x ) )
11:10:  |-  A. x ( ( z  e.  y  ->  ( y  e.  x  ->  z  e.  x ) )  <->  ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x ) )
12:1,11:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( z  e.  y  ->  ( y  e.  x  ->  z  e.  x ) )  <->  [. A  /  x ]. ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x ) ) ).
13:9,12:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ) ).
14:13:  |-  (. A  e.  B  ->.  A. y ( [. A  /  x ]. ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ) ).
15:14:  |-  (. A  e.  B  ->.  ( A. y [. A  /  x ]. ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ) ).
16:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. A. y ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  A. y [. A  /  x ]. ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x ) ) ).
17:15,16:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. A. y ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ) ).
18:17:  |-  (. A  e.  B  ->.  A. z ( [. A  /  x ]. A. y ( (  z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ) ).
19:18:  |-  (. A  e.  B  ->.  ( A. z [. A  /  x ]. A. y ( (  z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ) ).
20:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. A. z A. y ( (  z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  A. z [. A  /  x ]. A. y ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x ) ) ).
21:19,20:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. A. z A. y ( (  z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ) ).
22::  |-  ( Tr  A  <->  A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
23:21,22:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. A. z A. y ( (  z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  Tr  A ) ).
24::  |-  ( Tr  x  <->  A. z A. y ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x ) )
25:24:  |-  A. x ( Tr  x  <->  A. z A. y ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x ) )
26:1,25:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. Tr  x  <->  [. A  /  x ]. A. z A. y ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x ) ) ).
27:23,26:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. Tr  x  <->  Tr  A ) ).
qed:27:  |-  ( A  e.  B  ->  ( [. A  /  x ]. Tr  x  <->  Tr  A ) )
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  (
 [. A  /  x ].
 Tr  x  <->  Tr  A ) )
 
TheoremtruniALTVD 28844* The union of a class of transitive sets is transitive. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. truniALT 28481 is truniALTVD 28844 without virtual deductions and was automatically derived from truniALTVD 28844.
1::  |-  (. A. x  e.  A Tr  x  ->.  A. x  e.  A  Tr  x ).
2::  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A )  ->.  ( z  e.  y  /\  y  e.  U. A ) ).
3:2:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A )  ->.  z  e.  y ).
4:2:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A )  ->.  y  e.  U. A ).
5:4:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A )  ->.  E. q ( y  e.  q  /\  q  e.  A ) ).
6::  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A ) ,  ( y  e.  q  /\  q  e.  A )  ->.  ( y  e.  q  /\  q  e.  A ) ).
7:6:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A ) ,  ( y  e.  q  /\  q  e.  A )  ->.  y  e.  q ).
8:6:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A ) ,  ( y  e.  q  /\  q  e.  A )  ->.  q  e.  A ).
9:1,8:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A ) ,  ( y  e.  q  /\  q  e.  A )  ->.  [ q  /  x ] Tr  x ).
10:8,9:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A ) ,  ( y  e.  q  /\  q  e.  A )  ->.  Tr  q ).
11:3,7,10:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A ) ,  ( y  e.  q  /\  q  e.  A )  ->.  z  e.  q ).
12:11,8:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A ) ,  ( y  e.  q  /\  q  e.  A )  ->.  z  e.  U. A ).
13:12:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A )  ->.  ( ( y  e.  q  /\  q  e.  A )  ->  z  e.  U. A ) ).
14:13:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A )  ->.  A. q ( ( y  e.  q  /\  q  e.  A )  ->  z  e.  U. A ) ).
15:14:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A )  ->.  ( E. q ( y  e.  q  /\  q  e.  A )  ->  z  e.  U. A ) ).
16:5,15:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A )  ->.  z  e.  U. A ).
17:16:  |-  (. A. x  e.  A Tr  x  ->.  ( ( z  e.  y  /\  y  e.  U. A )  ->  z  e.  U. A ) ).
18:17:  |-  (. A. x  e.  A Tr  x  ->.  A. z A. y ( ( z  e.  y  /\  y  e.  U. A )  ->  z  e.  U. A ) ).
19:18:  |-  (. A. x  e.  A Tr  x  ->.  Tr  U. A ).
qed:19:  |-  ( A. x  e.  A Tr  x  ->  Tr  U. A )
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  e.  A  Tr  x  ->  Tr  U. A )
 
Theoremee33VD 28845 Non-virtual deduction form of e33 28700. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. ee33 28460 is ee33VD 28845 without virtual deductions and was automatically derived from ee33VD 28845.
h1::  |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
h2::  |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )
h3::  |-  ( th  ->  ( ta  ->  et ) )
4:1,3:  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  ->  et ) ) ) )
5:4:  |-  ( ta  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )
6:2,5:  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) ) )
7:6:  |-  ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) )
8:7:  |-  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )
qed:8:  |-  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) )
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )   &    |-  ( th  ->  ( ta  ->  et ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) )
 
TheoremtrintALTVD 28846* The intersection of a class of transitive sets is transitive. Virtual deduction proof of trintALT 28847. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. trintALT 28847 is trintALTVD 28846 without virtual deductions and was automatically derived from trintALTVD 28846.
1::  |-  (. A. x  e.  A Tr  x  ->.  A. x  e.  A Tr  x ).
2::  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A )  ->.  ( z  e.  y  /\  y  e.  |^| A ) ).
3:2:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A )  ->.  z  e.  y ).
4:2:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A )  ->.  y  e.  |^| A ).
5:4:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A )  ->.  A. q  e.  A y  e.  q ).
6:5:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A )  ->.  ( q  e.  A  ->  y  e.  q ) ).
7::  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A ) ,  q  e.  A  ->.  q  e.  A ).
8:7,6:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A ) ,  q  e.  A  ->.  y  e.  q ).
9:7,1:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A ) ,  q  e.  A  ->.  [ q  /  x ] Tr  x ).
10:7,9:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A ) ,  q  e.  A  ->.  Tr  q ).
11:10,3,8:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A ) ,  q  e.  A  ->.  z  e.  q ).
12:11:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A )  ->.  ( q  e.  A  ->  z  e.  q ) ).
13:12:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A )  ->.  A. q ( q  e.  A  ->  z  e.  q ) ).
14:13:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A )  ->.  A. q  e.  A z  e.  q ).
15:3,14:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A )  ->.  z  e.  |^| A ).
16:15:  |-  (. A. x  e.  A Tr  x  ->.  ( ( z  e.  y  /\  y  e.  |^| A )  ->  z  e.  |^| A ) ).
17:16:  |-  (. A. x  e.  A Tr  x  ->.  A. z A. y ( ( z  e.  y  /\  y  e.  |^| A )  ->  z  e.  |^| A ) ).
18:17:  |-  (. A. x  e.  A Tr  x  ->.  Tr  |^| A ).
qed:18:  |-  ( A. x  e.  A Tr  x  ->  Tr  |^| A )
(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  e.  A  Tr  x  ->  Tr  |^| A )
 
TheoremtrintALT 28847* The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 28847 is an alternative proof of trint 4309. trintALT 28847 is trintALTVD 28846 without virtual deductions and was automatically derived from trintALTVD 28846 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  e.  A  Tr  x  ->  Tr  |^| A )
 
Theoremundif3VD 28848 The first equality of Exercise 13 of [TakeutiZaring] p. 22. Virtual deduction proof of undif3 3594. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. undif3 3594 is undif3VD 28848 without virtual deductions and was automatically derived from undif3VD 28848.
1::  |-  ( x  e.  ( A  u.  ( B  \  C ) )  <->  ( x  e.  A  \/  x  e.  ( B  \  C ) ) )
2::  |-  ( x  e.  ( B  \  C )  <->  ( x  e.  B  /\  -.  x  e.  C ) )
3:2:  |-  ( ( x  e.  A  \/  x  e.  ( B  \  C ) )  <->  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) )
4:1,3:  |-  ( x  e.  ( A  u.  ( B  \  C ) )  <->  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) )
5::  |-  (. x  e.  A  ->.  x  e.  A ).
6:5:  |-  (. x  e.  A  ->.  ( x  e.  A  \/  x  e.  B ) ).
7:5:  |-  (. x  e.  A  ->.  ( -.  x  e.  C  \/  x  e.  A ) ).
8:6,7:  |-  (. x  e.  A  ->.  ( ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  \/  x  e.  A ) ) ).
9:8:  |-  ( x  e.  A  ->  ( ( x  e.  A  \/  x  e.  B )  /\  (  -.  x  e.  C  \/  x  e.  A ) ) )
10::  |-  (. ( x  e.  B  /\  -.  x  e.  C )  ->.  ( x  e.  B  /\  -.  x  e.  C ) ).
11:10:  |-  (. ( x  e.  B  /\  -.  x  e.  C )  ->.  x  e.  B ).
12:10:  |-  (. ( x  e.  B  /\  -.  x  e.  C )  ->.  -.  x  e.  C  ).
13:11:  |-  (. ( x  e.  B  /\  -.  x  e.  C )  ->.  ( x  e.  A  \/  x  e.  B ) ).
14:12:  |-  (. ( x  e.  B  /\  -.  x  e.  C )  ->.  ( -.  x  e.  C  \/  x  e.  A ) ).
15:13,14:  |-  (. ( x  e.  B  /\  -.  x  e.  C )  ->.  ( ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  \/  x  e.  A ) ) ).
16:15:  |-  ( ( x  e.  B  /\  -.  x  e.  C )  ->  ( ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  \/  x  e.  A ) ) )
17:9,16:  |-  ( ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) )  ->  ( ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  \/  x  e.  A ) ) )
18::  |-  (. ( x  e.  A  /\  -.  x  e.  C )  ->.  ( x  e.  A  /\  -.  x  e.  C ) ).
19:18:  |-  (. ( x  e.  A  /\  -.  x  e.  C )  ->.  x  e.  A ).
20:18:  |-  (. ( x  e.  A  /\  -.  x  e.  C )  ->.  -.  x  e.  C  ).
21:18:  |-  (. ( x  e.  A  /\  -.  x  e.  C )  ->.  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) ).
22:21:  |-  ( ( x  e.  A  /\  -.  x  e.  C )  ->  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) )
23::  |-  (. ( x  e.  A  /\  x  e.  A )  ->.  ( x  e.  A  /\  x  e.  A ) ).
24:23:  |-  (. ( x  e.  A  /\  x  e.  A )  ->.  x  e.  A ).
25:24:  |-  (. ( x  e.  A  /\  x  e.  A )  ->.  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) ).
26:25:  |-  ( ( x  e.  A  /\  x  e.  A )  ->  ( x  e.  A  \/  (  x  e.  B  /\  -.  x  e.  C ) ) )
27:10:  |-  (. ( x  e.  B  /\  -.  x  e.  C )  ->.  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) ).
28:27:  |-  ( ( x  e.  B  /\  -.  x  e.  C )  ->  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) )
29::  |-  (. ( x  e.  B  /\  x  e.  A )  ->.  ( x  e.  B  /\  x  e.  A ) ).
30:29:  |-  (. ( x  e.  B  /\  x  e.  A )  ->.  x  e.  A ).
31:30:  |-  (. ( x  e.  B  /\  x  e.  A )  ->.  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) ).
32:31:  |-  ( ( x  e.  B  /\  x  e.  A )  ->  ( x  e.  A  \/  (  x  e.  B  /\  -.  x  e.  C ) ) )
33:22,26:  |-  ( ( ( x  e.  A  /\  -.  x  e.  C )  \/  ( x  e.  A  /\  x  e.  A ) )  ->  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) )
34:28,32:  |-  ( ( ( x  e.  B  /\  -.  x  e.  C )  \/  ( x  e.  B  /\  x  e.  A ) )  ->  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) )
35:33,34:  |-  ( ( ( ( x  e.  A  /\  -.  x  e.  C )  \/  ( x  e.  A  /\  x  e.  A ) )  \/  ( ( x  e.  B  /\  -.  x  e.  C )  \/  ( x  e.  B  /\  x  e.  A ) ) )  ->  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) )
36::  |-  ( ( ( ( x  e.  A  /\  -.  x  e.  C )  \/  ( x  e.  A  /\  x  e.  A ) )  \/  ( ( x  e.  B  /\  -.  x  e.  C )  \/  ( x  e.  B  /\  x  e.  A ) ) )  <->  ( ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  \/  x  e.  A ) ) )
37:36,35:  |-  ( ( ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  \/  x  e.  A ) )  ->  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) )
38:17,37:  |-  ( ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) )  <->  ( ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  \/  x  e.  A ) ) )
39::  |-  ( x  e.  ( C  \  A )  <->  ( x  e.  C  /\  -.  x  e.  A ) )
40:39:  |-  ( -.  x  e.  ( C  \  A )  <->  -.  ( x  e.  C  /\  -.  x  e.  A ) )
41::  |-  ( -.  ( x  e.  C  /\  -.  x  e.  A )  <->  ( -.  x  e.  C  \/  x  e.  A ) )
42:40,41:  |-  ( -.  x  e.  ( C  \  A )  <->  ( -.  x  e.  C  \/  x  e.  A ) )
43::  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B  ) )
44:43,42:  |-  ( ( x  e.  ( A  u.  B )  /\  -.  x  e.  ( C  \  A )  )  <->  ( ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  /\  x  e.  A ) ) )
45::  |-  ( x  e.  ( ( A  u.  B )  \  ( C  \  A ) )  <->  (  x  e.  ( A  u.  B )  /\  -.  x  e.  ( C  \  A ) ) )
46:45,44:  |-  ( x  e.  ( ( A  u.  B )  \  ( C  \  A ) )  <->  (  ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  \/  x  e.  A ) ) )
47:4,38:  |-  ( x  e.  ( A  u.  ( B  \  C ) )  <->  ( ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  \/  x  e.  A ) ) )
48:46,47:  |-  ( x  e.  ( A  u.  ( B  \  C ) )  <->  x  e.  ( ( A  u.  B )  \  ( C  \  A ) ) )
49:48:  |-  A. x ( x  e.  ( A  u.  ( B  \  C ) )  <->  x  e.  ( ( A  u.  B )  \  ( C  \  A ) ) )
qed:49:  |-  ( A  u.  ( B  \  C ) )  =  ( ( A  u.  B )  \  ( C  \  A ) )
(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  u.  ( B  \  C ) )  =  ( ( A  u.  B )  \  ( C 
 \  A ) )
 
TheoremsbcssVD 28849 Virtual deduction proof of sbcss 3730. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcss 3730 is sbcssVD 28849 without virtual deductions and was automatically derived from sbcssVD 28849.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. y  e.  C  <->  y  e.  [_ A  /  x ]_ C ) ).
3:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. y  e.  D  <->  y  e.  [_ A  /  x ]_ D ) ).
4:2,3:  |-  (. A  e.  B  ->.  ( ( [. A  /  x ]. y  e.  C  ->  [. A  /  x ]. y  e.  D )  <->  ( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D  ) ) ).
5:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( y  e.  C  ->  y  e.  D )  <->  ( [. A  /  x ]. y  e.  C  ->  [. A  /  x ]. y  e.  D ) ) ).
6:4,5:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( y  e.  C  ->  y  e.  D )  <->  ( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D ) ) ).
7:6:  |-  (. A  e.  B  ->.  A. y ( [. A  /  x ]. ( y  e.  C  ->  y  e.  D )  <->  ( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D ) ) ).
8:7:  |-  (. A  e.  B  ->.  ( A. y [. A  /  x ]. ( y  e.  C  ->  y  e.  D )  <->  A. y ( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D )  ) ).
9:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. A. y ( y  e.  C  ->  y  e.  D )  <->  A. y [. A  /  x ]. ( y  e.  C  ->  y  e.  D ) ) ).
10:8,9:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. A. y ( y  e.  C  ->  y  e.  D )  <->  A. y ( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D )  ) ).
11::  |-  ( C  C_  D  <->  A. y ( y  e.  C  ->  y  e.  D ) )
110:11:  |-  A. x ( C  C_  D  <->  A. y ( y  e.  C  ->  y  e.  D ) )
12:1,110:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. C  C_  D  <->  [. A  /  x ]. A. y ( y  e.  C  ->  y  e.  D ) ) ).
13:10,12:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. C  C_  D  <->  A. y ( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D ) ) ).
14::  |-  ( [_ A  /  x ]_ C  C_  [_ A  /  x ]_ D  <->  A.  y ( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D ) )
15:13,14:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. C  C_  D  <->  [_ A  /  x ]_ C  C_  [_ A  /  x ]_ D ) ).
qed:15:  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  C_  D  <->  [_  A  /  x ]_ C  C_  [_ A  /  x ]_ D ) )
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  (
 [. A  /  x ]. C  C_  D  <->  [_ A  /  x ]_ C  C_  [_ A  /  x ]_ D ) )
 
TheoremcsbingVD 28850 Virtual deduction proof of csbing 3540. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbing 3540 is csbingVD 28850 without virtual deductions and was automatically derived from csbingVD 28850.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2::  |-  ( C  i^i  D )  =  { y  |  ( y  e.  C  /\  y  e.  D )  }
20:2:  |-  A. x ( C  i^i  D )  =  { y  |  ( y  e.  C  /\  y  e.  D ) }
30:1,20:  |-  (. A  e.  B  ->.  [. A  /  x ]. ( C  i^i  D )  =  { y  |  ( y  e.  C  /\  y  e.  D ) } ).
3:1,30:  |-  (. A  e.  B  ->.  [_ A  /  x ]_ ( C  i^i  D )  =  [_ A  /  x ]_ { y  |  ( y  e.  C  /\  y  e.  D ) } ).
4:1:  |-  (. A  e.  B  ->.  [_ A  /  x ]_ { y  |  ( y  e.  C  /\  y  e.  D ) }  =  { y  |  [. A  /  x ]. ( y  e.  C  /\  y  e.  D ) } ).
5:3,4:  |-  (. A  e.  B  ->.  [_ A  /  x ]_ ( C  i^i  D )  =  { y  |  [. A  /  x ]. ( y  e.  C  /\  y  e.  D ) } ).
6:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. y  e.  C  <->  y  e.  [_ A  /  x ]_ C ) ).
7:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. y  e.  D  <->  y  e.  [_ A  /  x ]_ D ) ).
8:6,7:  |-  (. A  e.  B  ->.  ( ( [. A  /  x ]. y  e.  C  /\  [. A  /  x ]. y  e.  D )  <->  ( y  e.  [_ A  /  x ]_ C  /\  y  e.  [_ A  /  x ]_ D )  ) ).
9:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( y  e.  C  /\  y  e.  D )  <->  ( [. A  /  x ]. y  e.  C  /\  [. A  /  x ]. y  e.  D ) ) ).
10:9,8:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( y  e.  C  /\  y  e.  D )  <->  ( y  e.  [_ A  /  x ]_ C  /\  y  e.  [_ A  /  x ]_ D ) ) ).
11:10:  |-  (. A  e.  B  ->.  A. y ( [. A  /  x ]. ( y  e.  C  /\  y  e.  D )  <->  ( y  e.  [_ A  /  x ]_ C  /\  y  e.  [_ A  /  x ]_ D ) ) ).
12:11:  |-  (. A  e.  B  ->.  { y  |  [. A  /  x ]. ( y  e.  C  /\  y  e.  D ) }  =  { y  |  ( y  e.  [_ A  /  x ]_ C  /\  y  e.  [_ A  /  x ]_ D ) } ).
13:5,12:  |-  (. A  e.  B  ->.  [_ A  /  x ]_ ( C  i^i  D )  =  { y  |  ( y  e.  [_ A  /  x ]_ C  /\  y  e.  [_ A  /  x ]_ D ) } ).
14::  |-  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D )  =  {  y  |  ( y  e.  [_ A  /  x ]_ C  /\  y  e.  [_ A  /  x ]_ D ) }
15:13,14:  |-  (. A  e.  B  ->.  [_ A  /  x ]_ ( C  i^i  D )  =  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) ).
qed:15:  |-  ( A  e.  B  ->  [_ A  /  x ]_ ( C  i^i  D )  =  (  [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) )
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  [_ A  /  x ]_ ( C  i^i  D )  =  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) )
 
TheoremonfrALTlem5VD 28851* Virtual deduction proof of onfrALTlem5 28483. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALTlem5 28483 is onfrALTlem5VD 28851 without virtual deductions and was automatically derived from onfrALTlem5VD 28851.
1::  |-  a  e.  _V
2:1:  |-  ( a  i^i  x )  e.  _V
3:2:  |-  ( [. ( a  i^i  x )  /  b ]. b  =  (/)  <->  ( a  i^i  x )  =  (/) )
4:3:  |-  ( -.  [. ( a  i^i  x )  /  b ]. b  =  (/)  <->  -.  ( a  i^i  x )  =  (/) )
5::  |-  ( ( a  i^i  x )  =/=  (/)  <->  -.  ( a  i^i  x  )  =  (/) )
6:4,5:  |-  ( -.  [. ( a  i^i  x )  /  b ]. b  =  (/)  <->  ( a  i^i  x )  =/=  (/) )
7:2:  |-  ( -.  [. ( a  i^i  x )  /  b ]. b  =  (/)  <->  [. ( a  i^i  x )  /  b ]. -.  b  =  (/) )
8::  |-  ( b  =/=  (/)  <->  -.  b  =  (/) )
9:8:  |-  A. b ( b  =/=  (/)  <->  -.  b  =  (/) )
10:2,9:  |-  ( [. ( a  i^i  x )  /  b ]. b  =/=  (/)  <->  [. ( a  i^i  x )  /  b ]. -.  b  =  (/) )
11:7,10:  |-  ( -.  [. ( a  i^i  x )  /  b ]. b  =  (/)  <->  [. ( a  i^i  x )  /  b ]. b  =/=  (/) )
12:6,11:  |-  ( [. ( a  i^i  x )  /  b ]. b  =/=  (/)  <->  (  a  i^i  x )  =/=  (/) )
13:2:  |-  ( [. ( a  i^i  x )  /  b ]. b  C_  ( a  i^i  x  )  <->  ( a  i^i  x )  C_  ( a  i^i  x ) )
14:12,13:  |-  ( ( [. ( a  i^i  x )  /  b ]. b  C_  ( a  i^i  x )  /\  [. ( a  i^i  x )  /  b ]. b  =/=  (/) )  <->  ( ( a  i^i  x )  C_  ( a  i^i  x )  /\  ( a  i^i  x )  =/=  (/) ) )
15:2:  |-  ( [. ( a  i^i  x )  /  b ]. ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  <->  ( [. ( a  i^i  x )  /  b ]. b  C_  ( a  i^i  x )  /\  [. ( a  i^i  x )  /  b ]. b  =/=  (/) ) )
16:15,14:  |-  ( [. ( a  i^i  x )  /  b ]. ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  <->  ( ( a  i^i  x )  C_  ( a  i^i  x )  /\  ( a  i^i  x )  =/=  (/) ) )
17:2:  |-  [_ ( a  i^i  x )  /  b ]_ ( b  i^i  y )  =  (  [_ ( a  i^i  x )  /  b ]_ b  i^i  [_ ( a  i^i  x )  /  b ]_ y )
18:2:  |-  [_ ( a  i^i  x )  /  b ]_ b  =  ( a  i^i  x )
19:2:  |-  [_ ( a  i^i  x )  /  b ]_ y  =  y
20:18,19:  |-  ( [_ ( a  i^i  x )  /  b ]_ b  i^i  [_ ( a  i^i  x )  /  b ]_ y )  =  ( ( a  i^i  x )  i^i  y )
21:17,20:  |-  [_ ( a  i^i  x )  /  b ]_ ( b  i^i  y )  =  ( (  a  i^i  x )  i^i  y )
22:2:  |-  ( [. ( a  i^i  x )  /  b ]. ( b  i^i  y )  =  (/)  <->  [_ ( a  i^i  x )  /  b ]_ ( b  i^i  y )  =  [_ ( a  i^i  x )  /  b ]_  (/) )
23:2:  |-  [_ ( a  i^i  x )  /  b ]_ (/)  =  (/)
24:21,23:  |-  ( [_ ( a  i^i  x )  /  b ]_ ( b  i^i  y )  =  [_ ( a  i^i  x )  /  b ]_ (/)  <->  ( ( a  i^i  x )  i^i  y )  =  (/) )
25:22,24:  |-  ( [. ( a  i^i  x )  /  b ]. ( b  i^i  y )  =  (/)  <->  ( ( a  i^i  x )  i^i  y )  =  (/) )
26:2:  |-  ( [. ( a  i^i  x )  /  b ]. y  e.  b  <->  y  e.  ( a  i^i  x ) )
27:25,26:  |-  ( ( [. ( a  i^i  x )  /  b ]. y  e.  b  /\  [.  ( a  i^i  x )  /  b ]. ( b  i^i  y )  =  (/) )  <->  ( y  e.  ( a  i^i  x )  /\  ( (  a  i^i  x )  i^i  y )  =  (/) ) )
28:2:  |-  ( [. ( a  i^i  x )  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  ( [. ( a  i^i  x )  /  b ]. y  e.  b  /\  [. ( a  i^i  x )  /  b ]. ( b  i^i  y )  =  (/) ) )
29:27,28:  |-  ( [. ( a  i^i  x )  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) )
30:29:  |-  A. y ( [. ( a  i^i  x )  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) )
31:30:  |-  ( E. y [. ( a  i^i  x )  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  E. y ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) )
32::  |-  ( E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y )  =  (/)  <->  E. y ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/)  ) )
33:31,32:  |-  ( E. y [. ( a  i^i  x )  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y )  =  (/) )
34:2:  |-  ( E. y [. ( a  i^i  x )  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  [. ( a  i^i  x )  /  b ]. E. y ( y  e.  b  /\  (  b  i^i  y )  =  (/) ) )
35:33,34:  |-  ( [. ( a  i^i  x )  /  b ]. E. y ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y  )  =  (/) )
36::  |-  ( E. y  e.  b ( b  i^i  y )  =  (/)  <->  E. y  ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) )
37:36:  |-  A. b ( E. y  e.  b ( b  i^i  y )  =  (/)  <->  E. y ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) )
38:2,37:  |-  ( [. ( a  i^i  x )  /  b ]. E. y  e.  b ( b  i^i  y )  =  (/)  <->  [. ( a  i^i  x )  /  b ]. E. y ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) )
39:35,38:  |-  ( [. ( a  i^i  x )  /  b ]. E. y  e.  b ( b  i^i  y )  =  (/)  <->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y )  =  (/) )
40:16,39:  |-  ( ( [. ( a  i^i  x )  /  b ]. ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  ->  [. ( a  i^i  x )  /  b ]. E. y  e.  b ( b  i^i  y )  =  (/) )  <->  ( ( ( a  i^i  x )  C_  ( a  i^i  x )  /\  ( a  i^i  x )  =/=  (/) )  ->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y )  =  (/) ) )
41:2:  |-  ( [. ( a  i^i  x )  /  b ]. ( ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  ->  E. y  e.  b ( b  i^i  y )  =  (/) )  <->  ( [. ( a  i^i  x )  /  b ]. ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  ->  [. ( a  i^i  x )  /  b ]. E. y  e.  b ( b  i^i  y )  =  (/) ) )
qed:40,41:  |-  ( [. ( a  i^i  x )  /  b ]. ( ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  ->  E. y  e.  b ( b  i^i  y )  =  (/) )  <->  ( ( ( a  i^i  x )  C_  ( a  i^i  x )  /\  ( a  i^i  x )  =/=  (/) )  ->  E. y  e.  ( a  i^i  x  ) ( ( a  i^i  x )  i^i  y )  =  (/) ) )
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( [. ( a  i^i  x )  /  b ]. (
 ( b  C_  (
 a  i^i  x )  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) )  <->  ( ( ( a  i^i  x ) 
 C_  ( a  i^i 
 x )  /\  (
 a  i^i  x )  =/= 
 (/) )  ->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y )  =  (/) ) )
 
TheoremonfrALTlem4VD 28852* Virtual deduction proof of onfrALTlem4 28484. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALTlem4 28484 is onfrALTlem4VD 28852 without virtual deductions and was automatically derived from onfrALTlem4VD 28852.
1::  |-  y  e.  _V
2:1:  |-  ( [. y  /  x ]. ( a  i^i  x )  =  (/)  <->  [_  y  /  x ]_ ( a  i^i  x )  =  [_ y  /  x ]_ (/) )
3:1:  |-  [_ y  /  x ]_ ( a  i^i  x )  =  ( [_ y  /  x ]_  a  i^i  [_ y  /  x ]_ x )
4:1:  |-  [_ y  /  x ]_ a  =  a
5:1:  |-  [_ y