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Theorem syl3anl2 1410
 Description: A syllogism inference. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 27-Jun-2022.)
Hypotheses
Ref Expression
syl3anl2.1 (𝜑𝜒)
syl3anl2.2 (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)
Assertion
Ref Expression
syl3anl2 (((𝜓𝜑𝜃) ∧ 𝜏) → 𝜂)

Proof of Theorem syl3anl2
StepHypRef Expression
1 syl3anl2.1 . . 3 (𝜑𝜒)
213anim2i 1150 . 2 ((𝜓𝜑𝜃) → (𝜓𝜒𝜃))
3 syl3anl2.2 . 2 (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)
42, 3sylan 583 1 (((𝜓𝜑𝜃) ∧ 𝜏) → 𝜂)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086 This theorem is referenced by:  chfacfscmulcl  21465  chfacfscmulgsum  21468  chfacfpmmulcl  21469  chfacfpmmulgsum  21472  cpmadumatpolylem1  21489  cpmadumatpolylem2  21490  cpmadumatpoly  21491  chcoeffeqlem  21493  2atlt  36728
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