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Theorem ee33VD 28333
Description: Non-virtual deduction form of e33 28188. 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 27949 is ee33VD 28333 without virtual deductions and was automatically derived from ee33VD 28333.
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 67 . . . . . 6  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  ->  et ) ) ) )
54com4r 82 . . . . 5  |-  ( ta 
->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )
61, 5syl8 67 . . . 4  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) ) )
7 pm2.43cbi 27945 . . . . 5  |-  ( (
ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) ) )  <-> 
( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) ) )
87biimpi 187 . . . 4  |-  ( (
ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) ) )  ->  ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) ) )
96, 8e0_ 28226 . . 3  |-  ( ps 
->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et )
) ) ) )
10 pm2.43cbi 27945 . . . 4  |-  ( ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) )  <->  ( ch  ->  (
ph  ->  ( ps  ->  ( ch  ->  et )
) ) ) )
1110biimpi 187 . . 3  |-  ( ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) )  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) )
129, 11e0_ 28226 . 2  |-  ( ch 
->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )
13 pm2.43cbi 27945 . . 3  |-  ( ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )
1413biimpi 187 . 2  |-  ( ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )  ->  ( ph  ->  ( ps  ->  ( ch  ->  et )
) ) )
1512, 14e0_ 28226 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 178
  Copyright terms: Public domain W3C validator