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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  381  jcad  514  a2and  844  zorn2lem6  10496  sqreulem  15306  ontopbas  35313  ontgval  35316  ordtoplem  35320  ordcmp  35332  fvineqsneu  36292  jaodd  41025  ee33  43282  sb5ALT  43286  tratrb  43297  onfrALTlem2  43307  onfrALT  43310  ax6e2ndeq  43320  ee22an  43434  sspwtrALT  43583  sspwtrALT2  43584  trintALT  43642
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