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Theorem simplbi2comgVD 28341
Description: Virtual deduction proof of simplbi2comg 1379. 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. simplbi2comg 1379 is simplbi2comgVD 28341 without virtual deductions and was automatically derived from simplbi2comgVD 28341.
1::  |-  (. ( ph  <->  ( ps  /\  ch ) )  ->.  ( ph  <->  (  ps  /\  ch ) ) ).
2:1:  |-  (. ( ph  <->  ( ps  /\  ch ) )  ->.  ( ( ps  /\  ch  )  ->  ph ) ).
3:2:  |-  (. ( ph  <->  ( ps  /\  ch ) )  ->.  ( ps  ->  ( ch  ->  ph ) ) ).
4:3:  |-  (. ( ph  <->  ( ps  /\  ch ) )  ->.  ( ch  ->  ( ps  ->  ph ) ) ).
qed:4:  |-  ( ( ph  <->  ( ps  /\  ch ) )  ->  ( ch  ->  ( ps  ->  ph ) ) )
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
simplbi2comgVD  |-  ( (
ph 
<->  ( ps  /\  ch ) )  ->  ( ch  ->  ( ps  ->  ph ) ) )

Proof of Theorem simplbi2comgVD
StepHypRef Expression
1 idn1 28006 . . . . 5  |-  (. ( ph 
<->  ( ps  /\  ch ) )  ->.  ( ph  <->  ( ps  /\  ch )
) ).
2 bi2 190 . . . . 5  |-  ( (
ph 
<->  ( ps  /\  ch ) )  ->  (
( ps  /\  ch )  ->  ph ) )
31, 2e1_ 28069 . . . 4  |-  (. ( ph 
<->  ( ps  /\  ch ) )  ->.  ( ( ps  /\  ch )  ->  ph ) ).
4 pm3.3 432 . . . 4  |-  ( ( ( ps  /\  ch )  ->  ph )  ->  ( ps  ->  ( ch  ->  ph ) ) )
53, 4e1_ 28069 . . 3  |-  (. ( ph 
<->  ( ps  /\  ch ) )  ->.  ( ps  ->  ( ch  ->  ph )
) ).
6 pm2.04 78 . . 3  |-  ( ( ps  ->  ( ch  ->  ph ) )  -> 
( ch  ->  ( ps  ->  ph ) ) )
75, 6e1_ 28069 . 2  |-  (. ( ph 
<->  ( ps  /\  ch ) )  ->.  ( ch  ->  ( ps  ->  ph )
) ).
87in1 28003 1  |-  ( (
ph 
<->  ( ps  /\  ch ) )  ->  ( ch  ->  ( ps  ->  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 178  df-an 361  df-vd1 28002
  Copyright terms: Public domain W3C validator