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Theorem syl6mpi 61
Description: A syllogism inference. (Contributed by Alan Sare, 8-Jul-2011.) (Proof shortened by Wolf Lammen, 13-Sep-2012.)
Hypotheses
Ref Expression
syl6mpi.1  |-  ( ph  ->  ( ps  ->  ch ) )
syl6mpi.2  |-  th
syl6mpi.3  |-  ( ch 
->  ( th  ->  ta ) )
Assertion
Ref Expression
syl6mpi  |-  ( ph  ->  ( ps  ->  ta ) )

Proof of Theorem syl6mpi
StepHypRef Expression
1 syl6mpi.1 . 2  |-  ( ph  ->  ( ps  ->  ch ) )
2 syl6mpi.2 . . 3  |-  th
3 syl6mpi.3 . . 3  |-  ( ch 
->  ( th  ->  ta ) )
42, 3mpi 20 . 2  |-  ( ch 
->  ta )
51, 4syl6 31 1  |-  ( ph  ->  ( ps  ->  ta ) )
Colors of variables: wff setvar 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:  19.8a  1881  suceloni  6631  bndrank  8291  ac10ct  8447  1re  9625  wl-19.8a  31392  tratrb  36327  ee20an  36550
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