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Theorem sylcom 28
Description: Syllogism inference with commutation of antecedents. (Contributed by NM, 29-Aug-2004.) (Proof shortened by O'Cat, 2-Feb-2006.) (Proof shortened by Stefan Allan, 23-Feb-2006.)
Hypotheses
Ref Expression
sylcom.1  |-  ( ph  ->  ( ps  ->  ch ) )
sylcom.2  |-  ( ps 
->  ( ch  ->  th )
)
Assertion
Ref Expression
sylcom  |-  ( ph  ->  ( ps  ->  th )
)

Proof of Theorem sylcom
StepHypRef Expression
1 sylcom.1 . 2  |-  ( ph  ->  ( ps  ->  ch ) )
2 sylcom.2 . . 3  |-  ( ps 
->  ( ch  ->  th )
)
32a2i 11 . 2  |-  ( ( ps  ->  ch )  ->  ( ps  ->  th )
)
41, 3syl 14 1  |-  ( ph  ->  ( ps  ->  th )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7
This theorem is referenced by:  syl5com  29  syl6  33  syli  37  mpbidi  151  stdcn  849  con4biddc  859  jaddc  866  con1biddc  878  necon4addc  2446  necon4bddc  2447  necon4ddc  2448  necon1addc  2452  necon1bddc  2453  dmcosseq  4950  iss  5005  funopg  5305  snon0  7037  metrest  14978
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