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Theorem syl312anc 1203
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 518 . 2  |-  ( ph  ->  ( et  /\  ze ) )
8 syl312anc.7 . 2  |-  ( ( ( ps  /\  ch  /\ 
th )  /\  ta  /\  ( et  /\  ze ) )  ->  si )
91, 2, 3, 4, 7, 8syl311anc 1196 1  |-  ( ph  ->  si )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934
This theorem is referenced by:  pythagtriplem19  12983  cdleme27cl  30624  cdlemefs27cl  30671  cdleme32fvcl  30698  cdlemg16ALTN  30916  cdlemg27a  30950  cdlemg31c  30957  cdlemg39  30974  cdlemk11ta  31187  cdlemk19ylem  31188  cdlemk11tc  31203  cdlemk45  31205  dihmeetlem12N  31577  dihjatc  31676
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
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