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Theorem sylnbi 636
Description: A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.)
Hypotheses
Ref Expression
sylnbi.1  |-  ( ph  <->  ps )
sylnbi.2  |-  ( -. 
ps  ->  ch )
Assertion
Ref Expression
sylnbi  |-  ( -. 
ph  ->  ch )

Proof of Theorem sylnbi
StepHypRef Expression
1 sylnbi.1 . . 3  |-  ( ph  <->  ps )
21notbii 627 . 2  |-  ( -. 
ph 
<->  -.  ps )
3 sylnbi.2 . 2  |-  ( -. 
ps  ->  ch )
42, 3sylbi 119 1  |-  ( -. 
ph  ->  ch )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  sylnbir  637  mo2n  1971  reuun2  3265  regexmidlem1  4312  iotanul  4949  riotaund  5581  snnen2og  6505
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