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Theorem eel0T1 28490
Description: An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
eel0T1.1  |-  ph
eel0T1.2  |-  (  T. 
->  ps )
eel0T1.3  |-  ( ch 
->  th )
eel0T1.4  |-  ( (
ph  /\  ps  /\  th )  ->  ta )
Assertion
Ref Expression
eel0T1  |-  ( ch 
->  ta )

Proof of Theorem eel0T1
StepHypRef Expression
1 3anass 938 . . 3  |-  ( (
ph  /\  T.  /\  ch ) 
<->  ( ph  /\  (  T.  /\  ch ) ) )
2 simpr 447 . . . 4  |-  ( (
ph  /\  (  T.  /\ 
ch ) )  -> 
(  T.  /\  ch ) )
3 eel0T1.1 . . . . 5  |-  ph
43jctl 525 . . . 4  |-  ( (  T.  /\  ch )  ->  ( ph  /\  (  T.  /\  ch ) ) )
52, 4impbii 180 . . 3  |-  ( (
ph  /\  (  T.  /\ 
ch ) )  <->  (  T.  /\ 
ch ) )
6 trcrm 24951 . . 3  |-  ( (  T.  /\  ch )  <->  ch )
71, 5, 63bitri 262 . 2  |-  ( (
ph  /\  T.  /\  ch ) 
<->  ch )
8 eel0T1.3 . . 3  |-  ( ch 
->  th )
9 eel0T1.2 . . . 4  |-  (  T. 
->  ps )
10 eel0T1.4 . . . 4  |-  ( (
ph  /\  ps  /\  th )  ->  ta )
119, 10syl3an2 1216 . . 3  |-  ( (
ph  /\  T.  /\  th )  ->  ta )
128, 11syl3an3 1217 . 2  |-  ( (
ph  /\  T.  /\  ch )  ->  ta )
137, 12sylbir 204 1  |-  ( ch 
->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    T. wtru 1307
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  df-an 360  df-3an 936  df-tru 1310
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