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Theorem syl7bi 223
Description: A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
syl7bi.1  |-  ( ph  <->  ps )
syl7bi.2  |-  ( ch 
->  ( th  ->  ( ps  ->  ta ) ) )
Assertion
Ref Expression
syl7bi  |-  ( ch 
->  ( th  ->  ( ph  ->  ta ) ) )

Proof of Theorem syl7bi
StepHypRef Expression
1 syl7bi.1 . . 3  |-  ( ph  <->  ps )
21biimpi 188 . 2  |-  ( ph  ->  ps )
3 syl7bi.2 . 2  |-  ( ch 
->  ( th  ->  ( ps  ->  ta ) ) )
42, 3syl7 66 1  |-  ( ch 
->  ( th  ->  ( ph  ->  ta ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178
This theorem is referenced by:  rspct  3046  zfpair  4402  gruen  8688  axpre-sup  9045  ndvdssub  12928  alexsubALT  18083  dfon2lem8  25418  prtlem15  26725  prtlem18  26727  fzofzim  28137
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 179
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