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Theorem syl223anc 1208
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 )
syl223anc.8  |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta )  /\  ( et  /\  ze  /\  si ) )  ->  rh )
Assertion
Ref Expression
syl223anc  |-  ( ph  ->  rh )

Proof of Theorem syl223anc
StepHypRef Expression
1 sylXanc.1 . 2  |-  ( ph  ->  ps )
2 sylXanc.2 . 2  |-  ( ph  ->  ch )
3 sylXanc.3 . . 3  |-  ( ph  ->  th )
4 sylXanc.4 . . 3  |-  ( ph  ->  ta )
53, 4jca 518 . 2  |-  ( ph  ->  ( th  /\  ta ) )
6 sylXanc.5 . 2  |-  ( ph  ->  et )
7 sylXanc.6 . 2  |-  ( ph  ->  ze )
8 sylXanc.7 . 2  |-  ( ph  ->  si )
9 syl223anc.8 . 2  |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta )  /\  ( et  /\  ze  /\  si ) )  ->  rh )
101, 2, 5, 6, 7, 8, 9syl213anc 1201 1  |-  ( ph  ->  rh )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934
This theorem is referenced by:  cdleme17d1  31100  cdlemednpq  31110  cdleme19d  31117  cdleme20aN  31120  cdleme20c  31122  cdleme20f  31125  cdleme20g  31126  cdleme20j  31129  cdleme20l1  31131  cdleme20l2  31132  cdlemky  31737  cdlemkyyN  31773
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
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