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Theorem syli 37
Description: Syllogism inference with common nested antecedent. (Contributed by NM, 4-Nov-2004.)
Hypotheses
Ref Expression
syli.1 (𝜓 → (𝜑𝜒))
syli.2 (𝜒 → (𝜑𝜃))
Assertion
Ref Expression
syli (𝜓 → (𝜑𝜃))

Proof of Theorem syli
StepHypRef Expression
1 syli.1 . 2 (𝜓 → (𝜑𝜒))
2 syli.2 . . 3 (𝜒 → (𝜑𝜃))
32com12 30 . 2 (𝜑 → (𝜒𝜃))
41, 3sylcom 28 1 (𝜓 → (𝜑𝜃))
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:  ibd  176  bijadc  810  sbi2v  1815  elab3gf  2751  elreldm  4608  tz6.12c  5255  rntpos  5926  smores  5961  f1domg  6326  negm  8833
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