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Theorem ee33VD 28655
Description: Non-virtual deduction form of e33 28509. 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. ee33 28284 is ee33VD 28655 without virtual deductions and was automatically derived from ee33VD 28655.
h1::  |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
h2::  |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )
h3::  |-  ( th  ->  ( ta  ->  et ) )
4:1,3:  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  ->  et ) ) ) )
5:4:  |-  ( ta  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )
6:2,5:  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) ) )
7:6:  |-  ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) )
8:7:  |-  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )
qed:8:  |-  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) )
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ee33VD.1  |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
ee33VD.2  |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )
ee33VD.3  |-  ( th 
->  ( ta  ->  et ) )
Assertion
Ref Expression
ee33VD  |-  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) )

Proof of Theorem ee33VD
StepHypRef Expression
1 ee33VD.2 . . . . 5  |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )
2 ee33VD.1 . . . . . . 7  |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
3 ee33VD.3 . . . . . . 7  |-  ( th 
->  ( ta  ->  et ) )
42, 3syl8 65 . . . . . 6  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  ->  et ) ) ) )
54com4r 80 . . . . 5  |-  ( ta 
->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )
61, 5syl8 65 . . . 4  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) ) )
7 pm2.43cbi 28280 . . . . 5  |-  ( (
ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) ) )  <-> 
( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) ) )
87biimpi 186 . . . 4  |-  ( (
ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) ) )  ->  ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) ) )
96, 8e0_ 28547 . . 3  |-  ( ps 
->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et )
) ) ) )
10 pm2.43cbi 28280 . . . 4  |-  ( ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) )  <->  ( ch  ->  (
ph  ->  ( ps  ->  ( ch  ->  et )
) ) ) )
1110biimpi 186 . . 3  |-  ( ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) )  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) )
129, 11e0_ 28547 . 2  |-  ( ch 
->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )
13 pm2.43cbi 28280 . . 3  |-  ( ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )
1413biimpi 186 . 2  |-  ( ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )  ->  ( ph  ->  ( ps  ->  ( ch  ->  et )
) ) )
1512, 14e0_ 28547 1  |-  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4
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 177
  Copyright terms: Public domain W3C validator