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Theorem simplbi2comgVD 28937
Description: Virtual deduction proof of simplbi2comg 1382. 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 1382 is simplbi2comgVD 28937 without virtual deductions and was automatically derived from simplbi2comgVD 28937.
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 28602 . . . . 5  |-  (. ( ph 
<->  ( ps  /\  ch ) )  ->.  ( ph  <->  ( ps  /\  ch )
) ).
2 bi2 190 . . . . 5  |-  ( (
ph 
<->  ( ps  /\  ch ) )  ->  (
( ps  /\  ch )  ->  ph ) )
31, 2e1_ 28665 . . . 4  |-  (. ( ph 
<->  ( ps  /\  ch ) )  ->.  ( ( ps  /\  ch )  ->  ph ) ).
4 pm3.3 432 . . . 4  |-  ( ( ( ps  /\  ch )  ->  ph )  ->  ( ps  ->  ( ch  ->  ph ) ) )
53, 4e1_ 28665 . . 3  |-  (. ( ph 
<->  ( ps  /\  ch ) )  ->.  ( ps  ->  ( ch  ->  ph )
) ).
6 pm2.04 78 . . 3  |-  ( ( ps  ->  ( ch  ->  ph ) )  -> 
( ch  ->  ( ps  ->  ph ) ) )
75, 6e1_ 28665 . 2  |-  (. ( ph 
<->  ( ps  /\  ch ) )  ->.  ( ch  ->  ( ps  ->  ph )
) ).
87in1 28599 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 28598
  Copyright terms: Public domain W3C validator