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Theorem syl7 69
Description: A syllogism rule of inference. The second premise is used to replace the third antecedent of the first premise. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)
Hypotheses
Ref Expression
syl7.1 (𝜑𝜓)
syl7.2 (𝜒 → (𝜃 → (𝜓𝜏)))
Assertion
Ref Expression
syl7 (𝜒 → (𝜃 → (𝜑𝜏)))

Proof of Theorem syl7
StepHypRef Expression
1 syl7.1 . . 3 (𝜑𝜓)
21a1i 9 . 2 (𝜒 → (𝜑𝜓))
3 syl7.2 . 2 (𝜒 → (𝜃 → (𝜓𝜏)))
42, 3syl5d 68 1 (𝜒 → (𝜃 → (𝜑𝜏)))
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:  syl7bi  164  const  837  syl3an3  1251  fvmptt  5512  nneneq  6751  pr2nelem  7047  ndvdssub  11634
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