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Theorem syl133anc 1224
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 )
syl133anc.8  |-  ( ( ps  /\  ( ch 
/\  th  /\  ta )  /\  ( et  /\  ze  /\  si ) )  ->  rh )
Assertion
Ref Expression
syl133anc  |-  ( ph  ->  rh )

Proof of Theorem syl133anc
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 1146 . 2  |-  ( ph  ->  ( et  /\  ze  /\  si ) )
9 syl133anc.8 . 2  |-  ( ( ps  /\  ( ch 
/\  th  /\  ta )  /\  ( et  /\  ze  /\  si ) )  ->  rh )
101, 2, 3, 4, 8, 9syl131anc 1214 1  |-  ( ph  ->  rh )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 947
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 949
This theorem is referenced by:  syl233anc  1230
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