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Theorem eelT01 28795
Description: An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
eelT01.1  |-  (  T. 
->  ph )
eelT01.2  |-  ps
eelT01.3  |-  ( ch 
->  th )
eelT01.4  |-  ( (
ph  /\  ps  /\  th )  ->  ta )
Assertion
Ref Expression
eelT01  |-  ( ch 
->  ta )

Proof of Theorem eelT01
StepHypRef Expression
1 3anass 938 . . 3  |-  ( (  T.  /\  ps  /\  ch )  <->  (  T.  /\  ( ps  /\  ch )
) )
2 trcrm 25054 . . 3  |-  ( (  T.  /\  ( ps 
/\  ch ) )  <->  ( ps  /\ 
ch ) )
3 simpr 447 . . . 4  |-  ( ( ps  /\  ch )  ->  ch )
4 eelT01.2 . . . . 5  |-  ps
54jctl 525 . . . 4  |-  ( ch 
->  ( ps  /\  ch ) )
63, 5impbii 180 . . 3  |-  ( ( ps  /\  ch )  <->  ch )
71, 2, 63bitri 262 . 2  |-  ( (  T.  /\  ps  /\  ch )  <->  ch )
8 eelT01.3 . . 3  |-  ( ch 
->  th )
9 eelT01.1 . . . 4  |-  (  T. 
->  ph )
10 eelT01.4 . . . 4  |-  ( (
ph  /\  ps  /\  th )  ->  ta )
119, 10syl3an1 1215 . . 3  |-  ( (  T.  /\  ps  /\  th )  ->  ta )
128, 11syl3an3 1217 . 2  |-  ( (  T.  /\  ps  /\  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
  Copyright terms: Public domain W3C validator