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Theorem 3syld 55
Description: Triple syllogism deduction. (Contributed by Jeff Hankins, 4-Aug-2009.)
Hypotheses
Ref Expression
3syld.1 (𝜑 → (𝜓𝜒))
3syld.2 (𝜑 → (𝜒𝜃))
3syld.3 (𝜑 → (𝜃𝜏))
Assertion
Ref Expression
3syld (𝜑 → (𝜓𝜏))

Proof of Theorem 3syld
StepHypRef Expression
1 3syld.1 . . 3 (𝜑 → (𝜓𝜒))
2 3syld.2 . . 3 (𝜑 → (𝜒𝜃))
31, 2syld 44 . 2 (𝜑 → (𝜓𝜃))
4 3syld.3 . 2 (𝜑 → (𝜃𝜏))
53, 4syld 44 1 (𝜑 → (𝜓𝜏))
Colors of variables: wff set class
Syntax hints:  wi 4
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7
This theorem is referenced by:  apreap  7652  msqge0  7681  cju  7989  facavg  9614  mulcn2  10064
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