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Theorem List for Metamath Proof Explorer - 28301-28400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremorbi1rVD 28301 Virtual deduction proof of orbi1r 27935. 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 28073  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ph )  ->.  ( ph  \/  ch ) ).
4:1,3,?: e12 28177  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ph )  ->.  ( ps  \/  ch ) ).
5:4,?: e2 28073  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ph )  ->.  ( ch  \/  ps ) ).
6:5:  |-  (. ( ph  <->  ps )  ->.  ( ( ch  \/  ph )  ->  ( ch  \/  ps ) ) ).
7::  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ch  \/  ps ) ).
8:7,?: e2 28073  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ps  \/  ch ) ).
9:1,8,?: e12 28177  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ph  \/  ch ) ).
10:9,?: e2 28073  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ch  \/  ph ) ).
11:10:  |-  (. ( ph  <->  ps )  ->.  ( ( ch  \/  ps )  ->  ( ch  \/  ph ) ) ).
12:6,11,?: e11 28130  |-  (. ( 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 28302 Virtual deduction proof of bitr3 27936. 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_ 28069  |-  (. ( ph  <->  ps )  ->.  ( ps  <->  ph ) ).
3::  |-  (. ( ph  <->  ps ) ,. ( ph  <->  ch )  ->.  ( ph  <->  ch ) ).
4:3,?: e2 28073  |-  (. ( ph  <->  ps ) ,. ( ph  <->  ch )  ->.  ( ch  <->  ph ) ).
5:2,4,?: e12 28177  |-  (. ( 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 28303 Virtual deduction proof of 3orbi123 27937. 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_ 28069  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ph  <->  ps ) ).
3:1,?: e1_ 28069  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ch  <->  th ) ).
4:1,?: e1_ 28069  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ta  <->  et ) ).
5:2,3,?: e11 28130  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ph  \/  ch )  <->  ( ps  \/  th ) ) ).
6:5,4,?: e11 28130  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ( ph  \/  ch )  \/  ta )  <->  ( ( ps  \/  th )  \/  et ) ) ).
7:?:  |-  ( ( ( ph  \/  ch )  \/  ta )  <->  ( ph  \/  ch  \/  ta ) )
8:6,7,?: e10 28136  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ph  \/  ch  \/  ta )  <->  ( ( ps  \/  th )  \/  et ) ) ).
9:?:  |-  ( ( ( ps  \/  th )  \/  et )  <->  ( ps  \/  th  \/  et ) )
10:8,9,?: e10 28136  |-  (. ( ( 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 28304 Virtual deduction proof of sbc3org 27959. 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_ 28069  |-  (. 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 28136  |-  (. A  e.  B  ->.  [. A  /  x ]. ( ( ( ph  \/  ps )  \/  ch )  <->  ( ph  \/  ps  \/  ch ) ) ).
4:1,33,?: e11 28130  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ( ph  \/  ps )  \/  ch )  <->  [. A  /  x ]. ( ph  \/  ps  \/  ch ) ) ).
5:2,4,?: e11 28130  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch ) ) ).
6:1,?: e1_ 28069  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) ).
7:6,?: e1_ 28069  |-  (. A  e.  B  ->.  ( ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) ).
8:5,7,?: e11 28130  |-  (. 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 28136  |-  (. 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 28305* Virtual deduction proof of alrim3con13v 27960. 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 28073  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ps ).
4:2,?: e2 28073  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ph ).
5:2,?: e2 28073  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ch ).
6:1,4,?: e12 28177  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ph ).
7:3,?: e2 28073  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ps ).
8:5,?: e2 28073  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ch ).
9:7,6,8,?: e222 28078  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ( A. x ps  /\  A. x ph  /\  A. x ch ) ).
10:9,?: e2 28073  |-  (. ( 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 28306 Virtual deduction proof of exbir 1371. 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 28177  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ,  ( ph  /\  ps )  ->.  ( ch  <->  th ) ).
6:3,5,?: e32 28211  |-  (. ( ( 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_ 28069  |-  (. ( ( 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 28307 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 28136  |-  (. ph  ->.  ( ps  ->  ( ch  <->  th ) ) ).
6:3,5,?: e21 28183  |-  (. ph ,. ps  ->.  ( ch  <->  th ) ).
7:4,6,?: e32 28211  |-  (. 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 28308* Virtual deduction proof of rspsbc2 27961. 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 28201  |-  (. 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 28201  |-  (. 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 28208  |-  (. 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 28309 Virtual deduction proof of 3impexp 1372. 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 28136  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ( ph  /\  ps )  /\  ch )  ->  th ) ).
4:3,?: e1_ 28069  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ph  /\  ps )  ->  ( ch  ->  th ) ) ).
5:4,?: e1_ 28069  |-  (. ( ( 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_ 28069  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ph  /\  ps )  ->  ( ch  ->  th ) ) ).
9:8,?: e1_ 28069  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ( ph  /\  ps )  /\  ch )  ->  th ) ).
10:2,9,?: e01 28133  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  th ) ).
11:10:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->  ( ( ph  /\  ps  /\  ch )  ->  th ) )
qed:6,11,?: e00 28221  |-  ( ( ( 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 28310 Virtual deduction proof of 3impexpbicom 1373. 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 28136  |-  (. ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  ( ta  <->  th ) ) ).
4:3,?: e1_ 28069  |-  (. ( ( 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_ 28069  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  ( ta  <->  th ) ) ).
8:7,2,?: e10 28136  |-  (. ( 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 28221  |-  ( ( ( 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 28311 Virtual deduction proof of 3impexpbicomi 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.
h1::  |-  ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )
qed:1,?: e0_ 28225  |-  ( 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 28312* Virtual deduction proof of sbcoreleleq 27962. 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_ 28069  |-  (. A  e.  B  ->.  ( [. A  /  y ]. x  e.  y  <->  x  e.  A ) ).
3:1,?: e1_ 28069  |-  (. A  e.  B  ->.  ( [. A  /  y ]. y  e.  x  <->  A  e.  x ) ).
4:1,?: e1_ 28069  |-  (. A  e.  B  ->.  ( [. A  /  y ]. x  =  y  <->  x  =  A ) ).
5:2,3,4,?: e111 28116  |-  (. 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_ 28069  |-  (. 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 28130  |-  (. 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 28313* Virtual deduction proof of nfra2 2703. 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 28221  |-  ( 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_ 28225  |-  ( 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 28221  |-  ( 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 28314* Virtual deduction proof of tratrb 27963. 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_ 28069  |-  (. ( 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_ 28069  |-  (. ( 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_ 28069  |-  (. ( 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 28073  |-  (. ( 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 28073  |-  (. ( 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 28098  |-  (. ( 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 28095  |-  (. ( 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 28077  |-  (. ( 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 28180  |-  (. ( 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 28208  |-  (. ( 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 28208  |-  (. ( 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 28180  |-  (. ( 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_ 28069  |-  (. ( 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 28098  |-  (. ( 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 28073  |-  (. ( 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 28177  |-  (. ( 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 28078  |-  (. ( 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_ 28225  |-  ( ( 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_ 28225  |-  ( ( 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_ 28069  |-  (. ( 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 28315 Virtual deduction proof of 3ax5 27964. 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_ 28069  |-  (. 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 28136  |-  (. 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 28316 Virtual deduction proof of syl5imp 27938. 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_ 28069  |-  (. ( ph  ->  ( ps  ->  ch ) )  ->.  ( ps  ->  ( ph  ->  ch ) ) ).
3::  |-  (. ( ph  ->  ( ps  ->  ch ) ) ,. ( th  ->  ps )  ->.  ( th  ->  ps ) ).
4:3,2,?: e21 28183  |-  (. ( ph  ->  ( ps  ->  ch ) ) ,. ( th  ->  ps )  ->.  ( th  ->  ( ph  ->  ch ) ) ).
5:4,?: e2 28073  |-  (. ( 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 28317 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_ 28225  |-  ph
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   =>    |-  ph
 
TheoremancomsimpVD 28318 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_ 28225  |-  ( ( ( 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 28319* Quantification restricted to a subclass for two quantifiers. ssralv 3350 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 27958 is ssralv2VD 28319 without virtual deductions and was automatically derived from ssralv2VD 28319.
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 28320 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 4544 using the Axiom of Regularity indirectly through dford2 7508. 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 27965 is ordelordALTVD 28320 without virtual deductions and was automatically derived from ordelordALTVD 28320 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 28321 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 3435 is equncomVD 28321 without virtual deductions and was automatically derived from equncomVD 28321.
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 28322 Inference form of equncom 3435. 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 3436 is equncomiVD 28322 without virtual deductions and was automatically derived from equncomiVD 28322.
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 28323 A set belongs to its successor. Alternate proof of sucid 4601. 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 28324 is sucidALTVD 28323 without virtual deductions and was automatically derived from sucidALTVD 28323. 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 4528, 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 7508.
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 28324 A set belongs to its successor. This proof was automatically derived from sucidALTVD 28323 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 28325 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 4601 is sucidVD 28325 without virtual deductions and was automatically derived from sucidVD 28325.
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 28326 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 27946 is imbi12VD 28326 without virtual deductions and was automatically derived from imbi12VD 28326.
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 28327 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 27947 is imbi13VD 28327 without virtual deductions and was automatically derived from imbi13VD 28327.
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 28328 Distribution of class substitution over a left-nested implication. Similar to sbcimg 3145. 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 27966 is sbcim2gVD 28328 without virtual deductions and was automatically derived from sbcim2gVD 28328.
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 28329 Implication form of sbcbiiOLD 3160. 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 27967 is sbcbiVD 28329 without virtual deductions and was automatically derived from sbcbiVD 28329.
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 28330* 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 27968 is trsbcVD 28330 without virtual deductions and was automatically derived from trsbcVD 28330.
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 28331* 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 27969 is truniALTVD 28331 without virtual deductions and was automatically derived from truniALTVD 28331.
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 28332 Non-virtual deduction form of e33 28187. 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 27948 is ee33VD 28332 without virtual deductions and was automatically derived from ee33VD 28332.
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 28333* The intersection of a class of transitive sets is transitive. Virtual deduction proof of trintALT 28334. 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 28334 is trintALTVD 28333 without virtual deductions and was automatically derived from trintALTVD 28333.
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 28334* The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 28334 is an alternative proof of trint 4258. trintALT 28334 is trintALTVD 28333 without virtual deductions and was automatically derived from trintALTVD 28333 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 28335 The first equality of Exercise 13 of [TakeutiZaring] p. 22. Virtual deduction proof of undif3 3545. 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 3545 is undif3VD 28335 without virtual deductions and was automatically derived from undif3VD 28335.
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 28336 Virtual deduction proof of sbcss 3681. 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 3681 is sbcssVD 28336 without virtual deductions and was automatically derived from sbcssVD 28336.
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 28337 Virtual deduction proof of csbing 3491. 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 3491 is csbingVD 28337 without virtual deductions and was automatically derived from csbingVD 28337.
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 28338* Virtual deduction proof of onfrALTlem5 27971. 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 27971 is onfrALTlem5VD 28338 without virtual deductions and was automatically derived from onfrALTlem5VD 28338.
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 28339* Virtual deduction proof of onfrALTlem4 27972. 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 27972 is onfrALTlem4VD 28339 without virtual deductions and was automatically derived from onfrALTlem4VD 28339.
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  /  x ]_ x  =  y
6:4,5:  |-  ( [_ y  /  x ]_ a  i^i  [_ y  /  x ]_ x )  =  (  a  i^i  y )
7:3,6:  |-  [_ y  /  x ]_ ( a  i^i  x )  =  ( a  i^i  y )
8:1:  |-  [_ y  /  x ]_ (/)  =  (/)
9:7,8:  |-  ( [_ y  /  x ]_ ( a  i^i  x )  =  [_ y  /  x ]_  (/)  <->  ( a  i^i  y )  =  (/) )
10:2,9:  |-  ( [. y  /  x ]. ( a  i^i  x )  =  (/)  <->  ( a  i^i  y )  =  (/) )
11:1:  |-  ( [. y  /  x ]. x  e.  a  <->  y  e.  a )
12:11,10:  |-  ( ( [. y  /  x ]. x  e.  a  /\  [. y  /  x ]. (  a  i^i  x )  =  (/) )  <->  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) )
13:1:  |-  ( [. y  /  x ]. ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  ( [. y  /  x ]. x  e.  a  /\  [. y  /  x ]. ( a  i^i  x )  =  (/) ) )
qed:13,12:  |-  ( [. y  /  x ]. ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) )
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( [. y  /  x ]. ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) )
 
TheoremonfrALTlem3VD 28340* Virtual deduction proof of onfrALTlem3 27973. 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. onfrALTlem3 27973 is onfrALTlem3VD 28340 without virtual deductions and was automatically derived from onfrALTlem3VD 28340.
1::  |-  (. ( a  C_  On  /\  a  =/=  (/) )  ->.  ( a  C_  On  /\  a  =/=  (/) ) ).
2::  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ).
3:2:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  x  e.  a ).
4:1:  |-  (. ( a  C_  On  /\  a  =/=  (/) )  ->.  a  C_  On ).
5:3,4:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  x  e.  On ).
6:5:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  Ord  x ).
7:6:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  _E  We  x ).
8::  |-  ( a  i^i  x )  C_  x
9:7,8:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  _E  We  ( a  i^i  x ) ).
10:9:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  _E  Fr  ( a  i^i  x ) ).
11:10:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  A. b ( ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  ->  E. y  e.  b ( b  i^i  y )  =  (/) ) ).
12::  |-  x  e.  _V
13:12,8:  |-  ( a  i^i  x )  e.  _V
14:13,11:  |-  (.