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Theorem syl23anc 1177
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 )
syl23anc.6  |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta  /\  et ) )  ->  ze )
Assertion
Ref Expression
syl23anc  |-  ( ph  ->  ze )

Proof of Theorem syl23anc
StepHypRef Expression
1 sylXanc.1 . . 3  |-  ( ph  ->  ps )
2 sylXanc.2 . . 3  |-  ( ph  ->  ch )
31, 2jca 300 . 2  |-  ( ph  ->  ( ps  /\  ch ) )
4 sylXanc.3 . 2  |-  ( ph  ->  th )
5 sylXanc.4 . 2  |-  ( ph  ->  ta )
6 sylXanc.5 . 2  |-  ( ph  ->  et )
7 syl23anc.6 . 2  |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta  /\  et ) )  ->  ze )
83, 4, 5, 6, 7syl13anc 1172 1  |-  ( ph  ->  ze )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115  df-3an 922
This theorem is referenced by:  gcdaddm  10755
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