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Theorem eelT11 28788
Description: An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
eelT11.1  |-  (  T. 
->  ph )
eelT11.2  |-  ( ps 
->  ch )
eelT11.3  |-  ( ps 
->  th )
eelT11.4  |-  ( (
ph  /\  ch  /\  th )  ->  ta )
Assertion
Ref Expression
eelT11  |-  ( ps 
->  ta )

Proof of Theorem eelT11
StepHypRef Expression
1 3anass 938 . . 3  |-  ( (  T.  /\  ps  /\  ps )  <->  (  T.  /\  ( ps  /\  ps )
) )
2 trcrm 25054 . . 3  |-  ( (  T.  /\  ( ps 
/\  ps ) )  <->  ( ps  /\ 
ps ) )
3 anidm 625 . . 3  |-  ( ( ps  /\  ps )  <->  ps )
41, 2, 33bitri 262 . 2  |-  ( (  T.  /\  ps  /\  ps )  <->  ps )
5 eelT11.3 . . 3  |-  ( ps 
->  th )
6 eelT11.2 . . . 4  |-  ( ps 
->  ch )
7 eelT11.1 . . . . 5  |-  (  T. 
->  ph )
8 eelT11.4 . . . . 5  |-  ( (
ph  /\  ch  /\  th )  ->  ta )
97, 8syl3an1 1215 . . . 4  |-  ( (  T.  /\  ch  /\  th )  ->  ta )
106, 9syl3an2 1216 . . 3  |-  ( (  T.  /\  ps  /\  th )  ->  ta )
115, 10syl3an3 1217 . 2  |-  ( (  T.  /\  ps  /\  ps )  ->  ta )
124, 11sylbir 204 1  |-  ( ps 
->  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