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Theorem syl6an 1364
Description: A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.)
Hypotheses
Ref Expression
syl6an.1  |-  ( ph  ->  ps )
syl6an.2  |-  ( ph  ->  ( ch  ->  th )
)
syl6an.3  |-  ( ( ps  /\  th )  ->  ta )
Assertion
Ref Expression
syl6an  |-  ( ph  ->  ( ch  ->  ta ) )

Proof of Theorem syl6an
StepHypRef Expression
1 syl6an.2 . . 3  |-  ( ph  ->  ( ch  ->  th )
)
2 syl6an.1 . . 3  |-  ( ph  ->  ps )
31, 2jctild 309 . 2  |-  ( ph  ->  ( ch  ->  ( ps  /\  th ) ) )
4 syl6an.3 . 2  |-  ( ( ps  /\  th )  ->  ta )
53, 4syl6 33 1  |-  ( ph  ->  ( ch  ->  ta ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia3 106
This theorem is referenced by:  prarloclem5  6788  ltsopr  6884  nominpos  8371  ublbneg  8815  absle  10160  rexanre  10291  rexico  10292  climshftlemg  10326  serif0  10374  dvds1lem  10398  dvds2lem  10399  bj-indind  10978
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