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Theorem sylsyld 58
Description: A double syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.)
Hypotheses
Ref Expression
sylsyld.1 (𝜑𝜓)
sylsyld.2 (𝜑 → (𝜒𝜃))
sylsyld.3 (𝜓 → (𝜃𝜏))
Assertion
Ref Expression
sylsyld (𝜑 → (𝜒𝜏))

Proof of Theorem sylsyld
StepHypRef Expression
1 sylsyld.2 . 2 (𝜑 → (𝜒𝜃))
2 sylsyld.1 . . 3 (𝜑𝜓)
3 sylsyld.3 . . 3 (𝜓 → (𝜃𝜏))
42, 3syl 14 . 2 (𝜑 → (𝜃𝜏))
51, 4syld 45 1 (𝜑 → (𝜒𝜏))
Colors of variables: wff set 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:  ax10o  1693  a16g  1836  trintssm  4042  funimaexglem  5206  smoiun  6198  findcard2  6783  ctssdc  6998  mkvprop  7032  ltexprlemrl  7425  archsr  7597  elfz0ubfz0  9909  ctinf  11950
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