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Theorem sylancb 646
Description: A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
Hypotheses
Ref Expression
sylancb.1  |-  ( ph  <->  ps )
sylancb.2  |-  ( ph  <->  ch )
sylancb.3  |-  ( ( ps  /\  ch )  ->  th )
Assertion
Ref Expression
sylancb  |-  ( ph  ->  th )

Proof of Theorem sylancb
StepHypRef Expression
1 sylancb.1 . . 3  |-  ( ph  <->  ps )
2 sylancb.2 . . 3  |-  ( ph  <->  ch )
3 sylancb.3 . . 3  |-  ( ( ps  /\  ch )  ->  th )
41, 2, 3syl2anb 465 . 2  |-  ( (
ph  /\  ph )  ->  th )
54anidms 626 1  |-  ( ph  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358
This theorem is referenced by:  svs2  25487
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
  Copyright terms: Public domain W3C validator