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Theorem ee233 27946
Description: Non-virtual deduction form of e233 28219. (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 31 . . . . . . . . . 10  |-  ( ph  ->  ( ps  ->  ( ta  ->  ( et  ->  ze ) ) ) )
65com3r 75 . . . . . . . . 9  |-  ( ta 
->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) )
72, 6syl8 67 . . . . . . . 8  |-  ( ph  ->  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) ) ) )
8 pm2.43cbi 27945 . . . . . . . 8  |-  ( (
ph  ->  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) ) ) )  <-> 
( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) ) ) )
97, 8mpbi 200 . . . . . . 7  |-  ( ps 
->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze )
) ) ) )
10 pm2.43cbi 27945 . . . . . . 7  |-  ( ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) ) )  <->  ( th  ->  (
ph  ->  ( ps  ->  ( et  ->  ze )
) ) ) )
119, 10mpbi 200 . . . . . 6  |-  ( th 
->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) )
1211com14 84 . . . . 5  |-  ( et 
->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) )
131, 12syl8 67 . . . 4  |-  ( ph  ->  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) ) ) )
14 pm2.43cbi 27945 . . . 4  |-  ( (
ph  ->  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) ) ) )  <-> 
( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) ) ) )
1513, 14mpbi 200 . . 3  |-  ( ps 
->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze )
) ) ) )
16 pm2.43cbi 27945 . . 3  |-  ( ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) ) )  <->  ( th  ->  (
ph  ->  ( ps  ->  ( th  ->  ze )
) ) ) )
1715, 16mpbi 200 . 2  |-  ( th 
->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) )
18 pm2.43cbi 27945 . 2  |-  ( ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) )  <->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) )
1917, 18mpbi 200 1  |-  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4
This theorem is referenced by:  truniALT  27970  onfrALTlem2  27976  e233  28219
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