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Theorem 3ax4VD 31953
Description: Virtual deduction proof of 3ax4 31598. 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,?: e1a 31704  |-  (. 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 31771  |-  (. 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.)
Assertion
Ref Expression
3ax4VD  |-  ( A. x ( ph  ->  ( ps  ->  ch )
)  ->  ( A. x ph  ->  ( A. x ps  ->  A. x ch ) ) )

Proof of Theorem 3ax4VD
StepHypRef Expression
1 idn1 31642 . . . 4  |-  (. A. x ( ph  ->  ( ps  ->  ch )
) 
->.  A. x ( ph  ->  ( ps  ->  ch ) ) ).
2 alim 1604 . . . 4  |-  ( A. x ( ph  ->  ( ps  ->  ch )
)  ->  ( A. x ph  ->  A. x
( ps  ->  ch ) ) )
31, 2e1a 31704 . . 3  |-  (. A. x ( ph  ->  ( ps  ->  ch )
) 
->.  ( A. x ph  ->  A. x ( ps 
->  ch ) ) ).
4 alim 1604 . . 3  |-  ( A. x ( ps  ->  ch )  ->  ( A. x ps  ->  A. x ch ) )
5 imim1 76 . . 3  |-  ( ( A. x ph  ->  A. x ( ps  ->  ch ) )  ->  (
( A. x ( ps  ->  ch )  ->  ( A. x ps 
->  A. x ch )
)  ->  ( A. x ph  ->  ( A. x ps  ->  A. x ch ) ) ) )
63, 4, 5e10 31771 . 2  |-  (. A. x ( ph  ->  ( ps  ->  ch )
) 
->.  ( A. x ph  ->  ( A. x ps 
->  A. x ch )
) ).
76in1 31639 1  |-  ( A. x ( ph  ->  ( ps  ->  ch )
)  ->  ( A. x ph  ->  ( A. x ps  ->  A. x ch ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-4 1603
This theorem depends on definitions:  df-bi 185  df-vd1 31638
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator