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Theorem eelTT1 28794
Description: An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
eelTT1.1  |-  (  T. 
->  ph )
eelTT1.2  |-  (  T. 
->  ps )
eelTT1.3  |-  ( ch 
->  th )
eelTT1.4  |-  ( (
ph  /\  ps  /\  th )  ->  ta )
Assertion
Ref Expression
eelTT1  |-  ( ch 
->  ta )

Proof of Theorem eelTT1
StepHypRef Expression
1 3anass 938 . . 3  |-  ( (  T.  /\  T.  /\  ch )  <->  (  T.  /\  (  T.  /\  ch )
) )
2 anabs5 784 . . 3  |-  ( (  T.  /\  (  T. 
/\  ch ) )  <->  (  T.  /\ 
ch ) )
3 trcrm 25054 . . 3  |-  ( (  T.  /\  ch )  <->  ch )
41, 2, 33bitri 262 . 2  |-  ( (  T.  /\  T.  /\  ch )  <->  ch )
5 eelTT1.3 . . 3  |-  ( ch 
->  th )
6 eelTT1.2 . . . 4  |-  (  T. 
->  ps )
7 eelTT1.1 . . . . 5  |-  (  T. 
->  ph )
8 eelTT1.4 . . . . 5  |-  ( (
ph  /\  ps  /\  th )  ->  ta )
97, 8syl3an1 1215 . . . 4  |-  ( (  T.  /\  ps  /\  th )  ->  ta )
106, 9syl3an2 1216 . . 3  |-  ( (  T.  /\  T.  /\  th )  ->  ta )
115, 10syl3an3 1217 . 2  |-  ( (  T.  /\  T.  /\  ch )  ->  ta )
124, 11sylbir 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