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Theorem syl7 68
Description: A syllogism rule of inference. The second premise is used to replace the third antecedent of the first premise. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)
Hypotheses
Ref Expression
syl7.1  |-  ( ph  ->  ps )
syl7.2  |-  ( ch 
->  ( th  ->  ( ps  ->  ta ) ) )
Assertion
Ref Expression
syl7  |-  ( ch 
->  ( th  ->  ( ph  ->  ta ) ) )

Proof of Theorem syl7
StepHypRef Expression
1 syl7.1 . . 3  |-  ( ph  ->  ps )
21a1i 9 . 2  |-  ( ch 
->  ( ph  ->  ps ) )
3 syl7.2 . 2  |-  ( ch 
->  ( th  ->  ( ps  ->  ta ) ) )
42, 3syl5d 67 1  |-  ( ch 
->  ( th  ->  ( ph  ->  ta ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7
This theorem is referenced by:  syl7bi  163  syl3an3  1205  fvmptt  5294  nneneq  6392  pr2nelem  6519  ndvdssub  10474
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