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Theorem simplbi2comtVD 33421
Description: Virtual deduction proof of simplbi2comt 626. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. simplbi2comt 626 is simplbi2comtVD 33421 without virtual deductions and was automatically derived from simplbi2comtVD 33421.
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
simplbi2comtVD  |-  ( (
ph 
<->  ( ps  /\  ch ) )  ->  ( ch  ->  ( ps  ->  ph ) ) )

Proof of Theorem simplbi2comtVD
StepHypRef Expression
1 idn1 33084 . . . . 5  |-  (. ( ph 
<->  ( ps  /\  ch ) )  ->.  ( ph  <->  ( ps  /\  ch )
) ).
2 bi2 198 . . . . 5  |-  ( (
ph 
<->  ( ps  /\  ch ) )  ->  (
( ps  /\  ch )  ->  ph ) )
31, 2e1a 33146 . . . 4  |-  (. ( ph 
<->  ( ps  /\  ch ) )  ->.  ( ( ps  /\  ch )  ->  ph ) ).
4 pm3.3 444 . . . 4  |-  ( ( ( ps  /\  ch )  ->  ph )  ->  ( ps  ->  ( ch  ->  ph ) ) )
53, 4e1a 33146 . . 3  |-  (. ( ph 
<->  ( ps  /\  ch ) )  ->.  ( ps  ->  ( ch  ->  ph )
) ).
6 pm2.04 82 . . 3  |-  ( ( ps  ->  ( ch  ->  ph ) )  -> 
( ch  ->  ( ps  ->  ph ) ) )
75, 6e1a 33146 . 2  |-  (. ( ph 
<->  ( ps  /\  ch ) )  ->.  ( ch  ->  ( ps  ->  ph )
) ).
87in1 33081 1  |-  ( (
ph 
<->  ( ps  /\  ch ) )  ->  ( ch  ->  ( ps  ->  ph ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-an 371  df-vd1 33080
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator