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Theorem syl321anc 1240
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 )
syl321anc.7  |-  ( ( ( ps  /\  ch  /\ 
th )  /\  ( ta  /\  et )  /\  ze )  ->  si )
Assertion
Ref Expression
syl321anc  |-  ( ph  ->  si )

Proof of Theorem syl321anc
StepHypRef Expression
1 sylXanc.1 . 2  |-  ( ph  ->  ps )
2 sylXanc.2 . 2  |-  ( ph  ->  ch )
3 sylXanc.3 . 2  |-  ( ph  ->  th )
4 sylXanc.4 . . 3  |-  ( ph  ->  ta )
5 sylXanc.5 . . 3  |-  ( ph  ->  et )
64, 5jca 532 . 2  |-  ( ph  ->  ( ta  /\  et ) )
7 sylXanc.6 . 2  |-  ( ph  ->  ze )
8 syl321anc.7 . 2  |-  ( ( ( ps  /\  ch  /\ 
th )  /\  ( ta  /\  et )  /\  ze )  ->  si )
91, 2, 3, 6, 7, 8syl311anc 1232 1  |-  ( ph  ->  si )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965
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 967
This theorem is referenced by:  syl322anc  1246  cxple2ad  22182  chordthmlem3  22241  lincext3  31002  4noncolr2  33110  4noncolr1  33111  3atlem5  33143  2lplnj  33276  llnmod2i2  33519  dalawlem11  33537  dalawlem12  33538  cdleme43dN  34148  cdleme4gfv  34163  cdlemeg46nlpq  34173  cdlemg17bq  34329  cdlemg31b0N  34350  cdlemg31b0a  34351  cdlemg31c  34355  cdlemg39  34372  cdlemk47  34605
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