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Theorem syl7bi 233
Description: A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 14-May-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 197 . 2  |-  ( ph  ->  ps )
3 syl7bi.2 . 2  |-  ( ch 
->  ( th  ->  ( ps  ->  ta ) ) )
42, 3syl7 70 1  |-  ( ch 
->  ( th  ->  ( ph  ->  ta ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 188
This theorem is referenced by:  rspct  3175  zfpair  4658  gruen  9245  axpre-sup  9601  nn0lt2  11008  fzofzim  11970  ndvdssub  14388  alexsubALT  21065  clwlkisclwwlklem2a  25512  erclwwlktr  25542  erclwwlkntr  25554  dfon2lem8  30444  prtlem15  32416  prtlem18  32418
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