Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  19.21a3con13vVD Unicode version

Theorem 19.21a3con13vVD 28628
Description: Virtual deduction proof of alrim3con13v 28296. 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 28403  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ps ).
4:2,?: e2 28403  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ph ).
5:2,?: e2 28403  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ch ).
6:1,4,?: e12 28499  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ph ).
7:3,?: e2 28403  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ps ).
8:5,?: e2 28403  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ch ).
9:7,6,8,?: e222 28408  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ( A. x ps  /\  A. x ph  /\  A. x ch ) ).
10:9,?: e2 28403  |-  (. ( 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.)
Assertion
Ref Expression
19.21a3con13vVD  |-  ( (
ph  ->  A. x ph )  ->  ( ( ps  /\  ph 
/\  ch )  ->  A. x
( ps  /\  ph  /\ 
ch ) ) )
Distinct variable groups:    ps, x    ch, x
Allowed substitution hint:    ph( x)

Proof of Theorem 19.21a3con13vVD
StepHypRef Expression
1 idn2 28385 . . . . . . 7  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\ 
ch )  ->.  ( ps  /\ 
ph  /\  ch ) ).
2 simp1 955 . . . . . . 7  |-  ( ( ps  /\  ph  /\  ch )  ->  ps )
31, 2e2 28403 . . . . . 6  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\ 
ch )  ->.  ps ).
4 ax-17 1603 . . . . . 6  |-  ( ps 
->  A. x ps )
53, 4e2 28403 . . . . 5  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\ 
ch )  ->.  A. x ps ).
6 idn1 28342 . . . . . 6  |-  (. ( ph  ->  A. x ph )  ->.  (
ph  ->  A. x ph ) ).
7 simp2 956 . . . . . . 7  |-  ( ( ps  /\  ph  /\  ch )  ->  ph )
81, 7e2 28403 . . . . . 6  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\ 
ch )  ->.  ph ).
9 id 19 . . . . . 6  |-  ( (
ph  ->  A. x ph )  ->  ( ph  ->  A. x ph ) )
106, 8, 9e12 28499 . . . . 5  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\ 
ch )  ->.  A. x ph ).
11 simp3 957 . . . . . . 7  |-  ( ( ps  /\  ph  /\  ch )  ->  ch )
121, 11e2 28403 . . . . . 6  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\ 
ch )  ->.  ch ).
13 ax-17 1603 . . . . . 6  |-  ( ch 
->  A. x ch )
1412, 13e2 28403 . . . . 5  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\ 
ch )  ->.  A. x ch ).
15 pm3.2an3 1131 . . . . 5  |-  ( A. x ps  ->  ( A. x ph  ->  ( A. x ch  ->  ( A. x ps  /\  A. x ph  /\  A. x ch ) ) ) )
165, 10, 14, 15e222 28408 . . . 4  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\ 
ch )  ->.  ( A. x ps  /\  A. x ph  /\  A. x ch ) ).
17 19.26-3an 1582 . . . . 5  |-  ( A. x ( ps  /\  ph 
/\  ch )  <->  ( A. x ps  /\  A. x ph  /\  A. x ch ) )
1817biimpri 197 . . . 4  |-  ( ( A. x ps  /\  A. x ph  /\  A. x ch )  ->  A. x
( ps  /\  ph  /\ 
ch ) )
1916, 18e2 28403 . . 3  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\ 
ch )  ->.  A. x
( ps  /\  ph  /\ 
ch ) ).
2019in2 28377 . 2  |-  (. ( ph  ->  A. x ph )  ->.  ( ( ps  /\  ph  /\ 
ch )  ->  A. x
( ps  /\  ph  /\ 
ch ) ) ).
2120in1 28339 1  |-  ( (
ph  ->  A. x ph )  ->  ( ( ps  /\  ph 
/\  ch )  ->  A. x
( ps  /\  ph  /\ 
ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934   A.wal 1527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-vd1 28338  df-vd2 28347
  Copyright terms: Public domain W3C validator