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Theorem eel12131 36964
Description: An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
Hypotheses
Ref Expression
eel12131.1  |-  ( ph  ->  ps )
eel12131.2  |-  ( (
ph  /\  ch )  ->  th )
eel12131.3  |-  ( (
ph  /\  ta )  ->  et )
eel12131.4  |-  ( ( ps  /\  th  /\  et )  ->  ze )
Assertion
Ref Expression
eel12131  |-  ( (
ph  /\  ch  /\  ta )  ->  ze )

Proof of Theorem eel12131
StepHypRef Expression
1 eel12131.3 . . . . . 6  |-  ( (
ph  /\  ta )  ->  et )
2 eel12131.2 . . . . . . . . . 10  |-  ( (
ph  /\  ch )  ->  th )
3 eel12131.1 . . . . . . . . . . 11  |-  ( ph  ->  ps )
4 eel12131.4 . . . . . . . . . . . 12  |-  ( ( ps  /\  th  /\  et )  ->  ze )
543exp 1205 . . . . . . . . . . 11  |-  ( ps 
->  ( th  ->  ( et  ->  ze ) ) )
63, 5syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( th  ->  ( et  ->  ze ) ) )
72, 6syl5com 32 . . . . . . . . 9  |-  ( (
ph  /\  ch )  ->  ( ph  ->  ( et  ->  ze ) ) )
87ex 436 . . . . . . . 8  |-  ( ph  ->  ( ch  ->  ( ph  ->  ( et  ->  ze ) ) ) )
98pm2.43b 53 . . . . . . 7  |-  ( ch 
->  ( ph  ->  ( et  ->  ze ) ) )
109com13 84 . . . . . 6  |-  ( et 
->  ( ph  ->  ( ch  ->  ze ) ) )
111, 10syl 17 . . . . 5  |-  ( (
ph  /\  ta )  ->  ( ph  ->  ( ch  ->  ze ) ) )
1211ex 436 . . . 4  |-  ( ph  ->  ( ta  ->  ( ph  ->  ( ch  ->  ze ) ) ) )
1312pm2.43b 53 . . 3  |-  ( ta 
->  ( ph  ->  ( ch  ->  ze ) ) )
1413com3l 85 . 2  |-  ( ph  ->  ( ch  ->  ( ta  ->  ze ) ) )
15143imp 1200 1  |-  ( (
ph  /\  ch  /\  ta )  ->  ze )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 983
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 189  df-an 373  df-3an 985
This theorem is referenced by:  isosctrlem1ALT  37198
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