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Theorem syl3c 62
Description: A syllogism inference combined with contraction. (Contributed by Alan Sare, 7-Jul-2011.)
Hypotheses
Ref Expression
syl3c.1  |-  ( ph  ->  ps )
syl3c.2  |-  ( ph  ->  ch )
syl3c.3  |-  ( ph  ->  th )
syl3c.4  |-  ( ps 
->  ( ch  ->  ( th  ->  ta ) ) )
Assertion
Ref Expression
syl3c  |-  ( ph  ->  ta )

Proof of Theorem syl3c
StepHypRef Expression
1 syl3c.3 . 2  |-  ( ph  ->  th )
2 syl3c.1 . . 3  |-  ( ph  ->  ps )
3 syl3c.2 . . 3  |-  ( ph  ->  ch )
4 syl3c.4 . . 3  |-  ( ps 
->  ( ch  ->  ( th  ->  ta ) ) )
52, 3, 4sylc 61 . 2  |-  ( ph  ->  ( th  ->  ta ) )
61, 5mpd 13 1  |-  ( 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:  bilukdc  1328  frirrg  4107  tfrlem1  5951  tfrcl  6007  caucvgprprlemval  6929  peano5uzti  8525  lcmneg  10589  prmind2  10635
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