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Theorem ee233 28281
Description: Non-virtual deduction form of e233 28540. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) 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.
h1::  |-  ( ph  ->  ( ps  ->  ch ) )
h2::  |-  ( ph  ->  ( ps  ->  ( th  ->  ta ) ) )
h3::  |-  ( ph  ->  ( ps  ->  ( th  ->  et ) ) )
h4::  |-  ( ch  ->  ( ta  ->  ( et  ->  ze ) ) )
5:1,4:  |-  ( ph  ->  ( ps  ->  ( ta  ->  ( et  ->  ze ) ) )  )
6:5:  |-  ( ta  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) )  )
7:2,6:  |-  ( ph  ->  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) ) ) )
8:7:  |-  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) ) )
9:8:  |-  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) )  )
10:9:  |-  ( ph  ->  ( ps  ->  ( th  ->  ( et  ->  ze ) ) )  )
11:10:  |-  ( et  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) )  )
12:3,11:  |-  ( ph  ->  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) ) ) )
13:12:  |-  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) ) )
14:13:  |-  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) )  )
qed:14:  |-  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) )
Hypotheses
Ref Expression
ee233.1  |-  ( ph  ->  ( ps  ->  ch ) )
ee233.2  |-  ( ph  ->  ( ps  ->  ( th  ->  ta ) ) )
ee233.3  |-  ( ph  ->  ( ps  ->  ( th  ->  et ) ) )
ee233.4  |-  ( ch 
->  ( ta  ->  ( et  ->  ze ) ) )
Assertion
Ref Expression
ee233  |-  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) )

Proof of Theorem ee233
StepHypRef Expression
1 ee233.3 . . . . 5  |-  ( ph  ->  ( ps  ->  ( th  ->  et ) ) )
2 ee233.2 . . . . . . . . 9  |-  ( ph  ->  ( ps  ->  ( th  ->  ta ) ) )
3 ee233.1 . . . . . . . . . . 11  |-  ( ph  ->  ( ps  ->  ch ) )
4 ee233.4 . . . . . . . . . . 11  |-  ( ch 
->  ( ta  ->  ( et  ->  ze ) ) )
53, 4syl6 29 . . . . . . . . . 10  |-  ( ph  ->  ( ps  ->  ( ta  ->  ( et  ->  ze ) ) ) )
65com3r 73 . . . . . . . . 9  |-  ( ta 
->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) )
72, 6syl8 65 . . . . . . . 8  |-  ( ph  ->  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) ) ) )
8 pm2.43cbi 28280 . . . . . . . 8  |-  ( (
ph  ->  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) ) ) )  <-> 
( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) ) ) )
97, 8mpbi 199 . . . . . . 7  |-  ( ps 
->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze )
) ) ) )
10 pm2.43cbi 28280 . . . . . . 7  |-  ( ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) ) )  <->  ( th  ->  (
ph  ->  ( ps  ->  ( et  ->  ze )
) ) ) )
119, 10mpbi 199 . . . . . 6  |-  ( th 
->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) )
1211com14 82 . . . . 5  |-  ( et 
->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) )
131, 12syl8 65 . . . 4  |-  ( ph  ->  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) ) ) )
14 pm2.43cbi 28280 . . . 4  |-  ( (
ph  ->  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) ) ) )  <-> 
( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) ) ) )
1513, 14mpbi 199 . . 3  |-  ( ps 
->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze )
) ) ) )
16 pm2.43cbi 28280 . . 3  |-  ( ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) ) )  <->  ( th  ->  (
ph  ->  ( ps  ->  ( th  ->  ze )
) ) ) )
1715, 16mpbi 199 . 2  |-  ( th 
->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) )
18 pm2.43cbi 28280 . 2  |-  ( ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) )  <->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) )
1917, 18mpbi 199 1  |-  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4
This theorem is referenced by:  truniALT  28305  onfrALTlem2  28311  e233  28540
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