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Theorem ee233 36919
Description: Non-virtual deduction form of e233 37191. (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 L. 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 34 . . . . . . . . . 10  |-  ( ph  ->  ( ps  ->  ( ta  ->  ( et  ->  ze ) ) ) )
65com3r 82 . . . . . . . . 9  |-  ( ta 
->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) )
72, 6syl8 72 . . . . . . . 8  |-  ( ph  ->  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) ) ) )
8 pm2.43cbi 36918 . . . . . . . 8  |-  ( (
ph  ->  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) ) ) )  <-> 
( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) ) ) )
97, 8mpbi 213 . . . . . . 7  |-  ( ps 
->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze )
) ) ) )
10 pm2.43cbi 36918 . . . . . . 7  |-  ( ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) ) )  <->  ( th  ->  (
ph  ->  ( ps  ->  ( et  ->  ze )
) ) ) )
119, 10mpbi 213 . . . . . 6  |-  ( th 
->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) )
1211com14 91 . . . . 5  |-  ( et 
->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) )
131, 12syl8 72 . . . 4  |-  ( ph  ->  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) ) ) )
14 pm2.43cbi 36918 . . . 4  |-  ( (
ph  ->  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) ) ) )  <-> 
( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) ) ) )
1513, 14mpbi 213 . . 3  |-  ( ps 
->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze )
) ) ) )
16 pm2.43cbi 36918 . . 3  |-  ( ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) ) )  <->  ( th  ->  (
ph  ->  ( ps  ->  ( th  ->  ze )
) ) ) )
1715, 16mpbi 213 . 2  |-  ( th 
->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) )
18 pm2.43cbi 36918 . 2  |-  ( ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) )  <->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) )
1917, 18mpbi 213 1  |-  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 190
This theorem is referenced by:  truniALT  36945  onfrALTlem2  36955  e233  37191
  Copyright terms: Public domain W3C validator