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Theorem syl221anc 1212
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 )
syl221anc.6  |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta )  /\  et )  ->  ze )
Assertion
Ref Expression
syl221anc  |-  ( ph  ->  ze )

Proof of Theorem syl221anc
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 304 . 2  |-  ( ph  ->  ( th  /\  ta ) )
6 sylXanc.5 . 2  |-  ( ph  ->  et )
7 syl221anc.6 . 2  |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta )  /\  et )  ->  ze )
81, 2, 5, 6, 7syl211anc 1207 1  |-  ( ph  ->  ze )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ 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:  syl222anc  1217  vtocldf  2711  dmdcanapd  8548  exprecap  10302  xrbdtri  11013  2strbasg  11987  2stropg  11988  cnptoprest  12335  blssps  12523  blss  12524  metequiv2  12592  xmettx  12606
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