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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  210  pm5.21ndd  379  jcad  512  a2and  846  zorn2lem6  10541  sqreulem  15398  ontopbas  36429  ontgval  36432  ordtoplem  36436  ordcmp  36448  fvineqsneu  37412  jaodd  42247  ee33  44541  sb5ALT  44545  tratrb  44556  onfrALTlem2  44566  onfrALT  44569  ax6e2ndeq  44579  ee22an  44693  sspwtrALT  44842  sspwtrALT2  44843  trintALT  44901
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