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Theorem eelT12 28792
Description: An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
eelT12.1  |-  (  T. 
->  ph )
eelT12.2  |-  ( ps 
->  ch )
eelT12.3  |-  ( th 
->  ta )
eelT12.4  |-  ( (
ph  /\  ch  /\  ta )  ->  et )
Assertion
Ref Expression
eelT12  |-  ( ( ps  /\  th )  ->  et )

Proof of Theorem eelT12
StepHypRef Expression
1 3anass 938 . . 3  |-  ( (  T.  /\  ps  /\  th )  <->  (  T.  /\  ( ps  /\  th )
) )
2 trcrm 25054 . . 3  |-  ( (  T.  /\  ( ps 
/\  th ) )  <->  ( ps  /\ 
th ) )
31, 2bitri 240 . 2  |-  ( (  T.  /\  ps  /\  th )  <->  ( ps  /\  th ) )
4 eelT12.3 . . 3  |-  ( th 
->  ta )
5 eelT12.2 . . . 4  |-  ( ps 
->  ch )
6 eelT12.1 . . . . 5  |-  (  T. 
->  ph )
7 eelT12.4 . . . . 5  |-  ( (
ph  /\  ch  /\  ta )  ->  et )
86, 7syl3an1 1215 . . . 4  |-  ( (  T.  /\  ch  /\  ta )  ->  et )
95, 8syl3an2 1216 . . 3  |-  ( (  T.  /\  ps  /\  ta )  ->  et )
104, 9syl3an3 1217 . 2  |-  ( (  T.  /\  ps  /\  th )  ->  et )
113, 10sylbir 204 1  |-  ( ( ps  /\  th )  ->  et )
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