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Theorem syl311anc 1234
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 )
syl311anc.6  |-  ( ( ( ps  /\  ch  /\ 
th )  /\  ta  /\  et )  ->  ze )
Assertion
Ref Expression
syl311anc  |-  ( ph  ->  ze )

Proof of Theorem syl311anc
StepHypRef Expression
1 sylXanc.1 . . 3  |-  ( ph  ->  ps )
2 sylXanc.2 . . 3  |-  ( ph  ->  ch )
3 sylXanc.3 . . 3  |-  ( ph  ->  th )
41, 2, 33jca 1162 . 2  |-  ( ph  ->  ( ps  /\  ch  /\ 
th ) )
5 sylXanc.4 . 2  |-  ( ph  ->  ta )
6 sylXanc.5 . 2  |-  ( ph  ->  et )
7 syl311anc.6 . 2  |-  ( ( ( ps  /\  ch  /\ 
th )  /\  ta  /\  et )  ->  ze )
84, 5, 6, 7syl3anc 1220 1  |-  ( ph  ->  ze )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116  df-3an 965
This theorem is referenced by:  syl312anc  1241  syl321anc  1242  syl313anc  1244  syl331anc  1245
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