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Theorem eelTTT 28786
Description: An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
eelTTT.1  |-  (  T. 
->  ph )
eelTTT.2  |-  (  T. 
->  ps )
eelTTT.3  |-  (  T. 
->  ch )
eelTTT.4  |-  ( (
ph  /\  ps  /\  ch )  ->  th )
Assertion
Ref Expression
eelTTT  |-  th

Proof of Theorem eelTTT
StepHypRef Expression
1 eelTTT.3 . . 3  |-  (  T. 
->  ch )
2 trcrm 25054 . . . 4  |-  ( (  T.  /\  ch )  <->  ch )
3 eelTTT.2 . . . . 5  |-  (  T. 
->  ps )
4 3anass 938 . . . . . . 7  |-  ( (  T.  /\  ps  /\  ch )  <->  (  T.  /\  ( ps  /\  ch )
) )
5 trcrm 25054 . . . . . . 7  |-  ( (  T.  /\  ( ps 
/\  ch ) )  <->  ( ps  /\ 
ch ) )
64, 5bitri 240 . . . . . 6  |-  ( (  T.  /\  ps  /\  ch )  <->  ( ps  /\  ch ) )
7 eelTTT.1 . . . . . . 7  |-  (  T. 
->  ph )
8 eelTTT.4 . . . . . . 7  |-  ( (
ph  /\  ps  /\  ch )  ->  th )
97, 8syl3an1 1215 . . . . . 6  |-  ( (  T.  /\  ps  /\  ch )  ->  th )
106, 9sylbir 204 . . . . 5  |-  ( ( ps  /\  ch )  ->  th )
113, 10sylan 457 . . . 4  |-  ( (  T.  /\  ch )  ->  th )
122, 11sylbir 204 . . 3  |-  ( ch 
->  th )
131, 12syl 15 . 2  |-  (  T. 
->  th )
1413trud 1314 1  |-  th
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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