MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  syl213anc Unicode version

Theorem syl213anc 1201
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 )
syl213anc.7  |-  ( ( ( ps  /\  ch )  /\  th  /\  ( ta  /\  et  /\  ze ) )  ->  si )
Assertion
Ref Expression
syl213anc  |-  ( ph  ->  si )

Proof of Theorem syl213anc
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 syl213anc.7 . 2  |-  ( ( ( ps  /\  ch )  /\  th  /\  ( ta  /\  et  /\  ze ) )  ->  si )
93, 4, 5, 6, 7, 8syl113anc 1194 1  |-  ( ph  ->  si )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934
This theorem is referenced by:  syl223anc  1208
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