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Theorem syl312anc 1249
Description: Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
Hypotheses
Ref Expression
sylXanc.1  |-  ( ph  ->  ps )
sylXanc.2  |-  ( ph  ->  ch )
sylXanc.3  |-  ( ph  ->  th )
sylXanc.4  |-  ( ph  ->  ta )
sylXanc.5  |-  ( ph  ->  et )
sylXanc.6  |-  ( ph  ->  ze )
syl312anc.7  |-  ( ( ( ps  /\  ch  /\ 
th )  /\  ta  /\  ( et  /\  ze ) )  ->  si )
Assertion
Ref Expression
syl312anc  |-  ( ph  ->  si )

Proof of Theorem syl312anc
StepHypRef Expression
1 sylXanc.1 . 2  |-  ( ph  ->  ps )
2 sylXanc.2 . 2  |-  ( ph  ->  ch )
3 sylXanc.3 . 2  |-  ( ph  ->  th )
4 sylXanc.4 . 2  |-  ( ph  ->  ta )
5 sylXanc.5 . . 3  |-  ( ph  ->  et )
6 sylXanc.6 . . 3  |-  ( ph  ->  ze )
75, 6jca 532 . 2  |-  ( ph  ->  ( et  /\  ze ) )
8 syl312anc.7 . 2  |-  ( ( ( ps  /\  ch  /\ 
th )  /\  ta  /\  ( et  /\  ze ) )  ->  si )
91, 2, 3, 4, 7, 8syl311anc 1242 1  |-  ( ph  ->  si )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975
This theorem is referenced by:  pythagtriplem19  14212  cdleme27cl  35162  cdlemefs27cl  35209  cdleme32fvcl  35236  cdlemg16ALTN  35454  cdlemg27a  35488  cdlemg31c  35495  cdlemg39  35512  cdlemk11ta  35725  cdlemk19ylem  35726  cdlemk11tc  35741  cdlemk45  35743  dihmeetlem12N  36115  dihjatc  36214
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