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Theorem com3rgbi 28308
Description: 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. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
1::  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->  ( ph  ->  ( ch  ->  ( ps  ->  th ) ) ) )
2::  |-  ( ( ph  ->  ( ch  ->  ( ps  ->  th ) ) )  ->  ( ch  ->  ( ph  ->  ( ps  ->  th ) ) ) )
3:1,2:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->  ( ch  ->  ( ph  ->  ( ps  ->  th ) ) ) )
4::  |-  ( ( ch  ->  ( ph  ->  ( ps  ->  th ) ) )  ->  ( ph  ->  ( ch  ->  ( ps  ->  th ) ) ) )
5::  |-  ( ( ph  ->  ( ch  ->  ( ps  ->  th ) ) )  ->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
6:4,5:  |-  ( ( ch  ->  ( ph  ->  ( ps  ->  th ) ) )  ->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
qed:3,6:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  <->  ( ch  ->  ( ph  ->  ( ps  ->  th ) ) ) )
Assertion
Ref Expression
com3rgbi  |-  ( (
ph  ->  ( ps  ->  ( ch  ->  th )
) )  <->  ( ch  ->  ( ph  ->  ( ps  ->  th ) ) ) )

Proof of Theorem com3rgbi
StepHypRef Expression
1 pm2.04 78 . . 3  |-  ( (
ph  ->  ( ps  ->  ( ch  ->  th )
) )  ->  ( ps  ->  ( ph  ->  ( ch  ->  th )
) ) )
21com24 83 . 2  |-  ( (
ph  ->  ( ps  ->  ( ch  ->  th )
) )  ->  ( ch  ->  ( ph  ->  ( ps  ->  th )
) ) )
3 pm2.04 78 . . 3  |-  ( ( ch  ->  ( ph  ->  ( ps  ->  th )
) )  ->  ( ph  ->  ( ch  ->  ( ps  ->  th )
) ) )
43com34 79 . 2  |-  ( ( ch  ->  ( ph  ->  ( ps  ->  th )
) )  ->  ( ph  ->  ( ps  ->  ( ch  ->  th )
) ) )
52, 4impbii 181 1  |-  ( (
ph  ->  ( ps  ->  ( ch  ->  th )
) )  <->  ( ch  ->  ( ph  ->  ( ps  ->  th ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177
This theorem is referenced by:  impexp3acom3r  28309
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
  Copyright terms: Public domain W3C validator