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Theorem syl6c 70
Description: Inference combining syl6 35 with contraction. (Contributed by Alan Sare, 2-May-2011.)
Hypotheses
Ref Expression
syl6c.1 (𝜑 → (𝜓𝜒))
syl6c.2 (𝜑 → (𝜓𝜃))
syl6c.3 (𝜒 → (𝜃𝜏))
Assertion
Ref Expression
syl6c (𝜑 → (𝜓𝜏))

Proof of Theorem syl6c
StepHypRef Expression
1 syl6c.2 . 2 (𝜑 → (𝜓𝜃))
2 syl6c.1 . . 3 (𝜑 → (𝜓𝜒))
3 syl6c.3 . . 3 (𝜒 → (𝜃𝜏))
42, 3syl6 35 . 2 (𝜑 → (𝜓 → (𝜃𝜏)))
51, 4mpdd 43 1 (𝜑 → (𝜓𝜏))
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:  syl6ci  71  syldd  72  impbidd  209  pm5.21ndd  378  jcad  511  a2and  843  zorn2lem6  10544  sqreulem  15364  ontopbas  36140  ontgval  36143  ordtoplem  36147  ordcmp  36159  fvineqsneu  37118  jaodd  41931  ee33  44197  sb5ALT  44201  tratrb  44212  onfrALTlem2  44222  onfrALT  44225  ax6e2ndeq  44235  ee22an  44349  sspwtrALT  44498  sspwtrALT2  44499  trintALT  44557
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