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Theorem ee223 31658
Description: e223 31659 without virtual deductions. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ee223.1  |-  ( ph  ->  ( ps  ->  ch ) )
ee223.2  |-  ( ph  ->  ( ps  ->  th )
)
ee223.3  |-  ( ph  ->  ( ps  ->  ( ta  ->  et ) ) )
ee223.4  |-  ( ch 
->  ( th  ->  ( et  ->  ze ) ) )
Assertion
Ref Expression
ee223  |-  ( ph  ->  ( ps  ->  ( ta  ->  ze ) ) )

Proof of Theorem ee223
StepHypRef Expression
1 ee223.2 . 2  |-  ( ph  ->  ( ps  ->  th )
)
2 ee223.3 . . . . . . . 8  |-  ( ph  ->  ( ps  ->  ( ta  ->  et ) ) )
3 ee223.1 . . . . . . . . . . . 12  |-  ( ph  ->  ( ps  ->  ch ) )
4 ee223.4 . . . . . . . . . . . 12  |-  ( ch 
->  ( th  ->  ( et  ->  ze ) ) )
53, 4syl6 33 . . . . . . . . . . 11  |-  ( ph  ->  ( ps  ->  ( th  ->  ( et  ->  ze ) ) ) )
65com34 83 . . . . . . . . . 10  |-  ( ph  ->  ( ps  ->  ( et  ->  ( th  ->  ze ) ) ) )
76com23 78 . . . . . . . . 9  |-  ( ph  ->  ( et  ->  ( ps  ->  ( th  ->  ze ) ) ) )
87com12 31 . . . . . . . 8  |-  ( et 
->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) )
92, 8syl8 70 . . . . . . 7  |-  ( ph  ->  ( ps  ->  ( ta  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) ) ) )
109com34 83 . . . . . 6  |-  ( ph  ->  ( ps  ->  ( ph  ->  ( ta  ->  ( ps  ->  ( th  ->  ze ) ) ) ) ) )
1110pm2.43a 49 . . . . 5  |-  ( ph  ->  ( ps  ->  ( ta  ->  ( ps  ->  ( th  ->  ze )
) ) ) )
1211com34 83 . . . 4  |-  ( ph  ->  ( ps  ->  ( ps  ->  ( ta  ->  ( th  ->  ze )
) ) ) )
1312pm2.43d 48 . . 3  |-  ( ph  ->  ( ps  ->  ( ta  ->  ( th  ->  ze ) ) ) )
1413com34 83 . 2  |-  ( ph  ->  ( ps  ->  ( th  ->  ( ta  ->  ze ) ) ) )
151, 14mpdd 40 1  |-  ( ph  ->  ( ps  ->  ( ta  ->  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
This theorem is referenced by:  e223  31659
  Copyright terms: Public domain W3C validator