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Theorem syl313anc 1206
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-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 )
sylXanc.7  |-  ( ph  ->  si )
syl313anc.8  |-  ( ( ( ps  /\  ch  /\ 
th )  /\  ta  /\  ( et  /\  ze  /\  si ) )  ->  rh )
Assertion
Ref Expression
syl313anc  |-  ( ph  ->  rh )

Proof of Theorem syl313anc
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 )
7 sylXanc.7 . . 3  |-  ( ph  ->  si )
85, 6, 73jca 1132 . 2  |-  ( ph  ->  ( et  /\  ze  /\  si ) )
9 syl313anc.8 . 2  |-  ( ( ( ps  /\  ch  /\ 
th )  /\  ta  /\  ( et  /\  ze  /\  si ) )  ->  rh )
101, 2, 3, 4, 8, 9syl311anc 1196 1  |-  ( ph  ->  rh )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934
This theorem is referenced by:  syl323anc  1212  osumcllem6N  30772  cdlemg13  31463  cdlemk7u  31681  cdlemk31  31707  cdlemk27-3  31718  cdlemk19ylem  31741  cdlemk46  31759
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
  Copyright terms: Public domain W3C validator