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Theorem sylan2d 288
Description: A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
Hypotheses
Ref Expression
sylan2d.1  |-  ( ph  ->  ( ps  ->  ch ) )
sylan2d.2  |-  ( ph  ->  ( ( th  /\  ch )  ->  ta )
)
Assertion
Ref Expression
sylan2d  |-  ( ph  ->  ( ( th  /\  ps )  ->  ta )
)

Proof of Theorem sylan2d
StepHypRef Expression
1 sylan2d.1 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
2 sylan2d.2 . . . 4  |-  ( ph  ->  ( ( th  /\  ch )  ->  ta )
)
32ancomsd 265 . . 3  |-  ( ph  ->  ( ( ch  /\  th )  ->  ta )
)
41, 3syland 287 . 2  |-  ( ph  ->  ( ( ps  /\  th )  ->  ta )
)
54ancomsd 265 1  |-  ( ph  ->  ( ( th  /\  ps )  ->  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-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  syl2and  289  sylan2i  399  swopo  4097  prarloclemlo  6956  prodgt02  8208  prodge02  8210
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