MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  syl233anc Unicode version

Theorem syl233anc 1211
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 )
sylXanc.8  |-  ( ph  ->  rh )
syl233anc.9  |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta  /\  et )  /\  ( ze  /\  si  /\  rh ) )  ->  mu )
Assertion
Ref Expression
syl233anc  |-  ( ph  ->  mu )

Proof of Theorem syl233anc
StepHypRef Expression
1 sylXanc.1 . . 3  |-  ( ph  ->  ps )
2 sylXanc.2 . . 3  |-  ( ph  ->  ch )
31, 2jca 518 . 2  |-  ( ph  ->  ( ps  /\  ch ) )
4 sylXanc.3 . 2  |-  ( ph  ->  th )
5 sylXanc.4 . 2  |-  ( ph  ->  ta )
6 sylXanc.5 . 2  |-  ( ph  ->  et )
7 sylXanc.6 . 2  |-  ( ph  ->  ze )
8 sylXanc.7 . 2  |-  ( ph  ->  si )
9 sylXanc.8 . 2  |-  ( ph  ->  rh )
10 syl233anc.9 . 2  |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta  /\  et )  /\  ( ze  /\  si  /\  rh ) )  ->  mu )
113, 4, 5, 6, 7, 8, 9, 10syl133anc 1205 1  |-  ( ph  ->  mu )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934
This theorem is referenced by:  2llnjN  30378  cdleme16b  31090  cdleme18d  31106  cdleme19d  31117  cdleme20bN  31121  cdleme20l1  31131  cdleme22cN  31153  cdleme22eALTN  31156  cdleme22f  31157  cdlemg33c0  31513  cdlemk5  31647  cdlemk5u  31672  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
  Copyright terms: Public domain W3C validator