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Theorem syl3anl2 1367
 Description: A syllogism inference. (Contributed by NM, 24-Feb-2005.)
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)
2 syl3anl2.2 . . . 4 (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)
32ex 449 . . 3 ((𝜓𝜒𝜃) → (𝜏𝜂))
41, 3syl3an2 1352 . 2 ((𝜓𝜑𝜃) → (𝜏𝜂))
54imp 444 1 (((𝜓𝜑𝜃) ∧ 𝜏) → 𝜂)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031 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 196  df-an 385  df-3an 1033 This theorem is referenced by:  syl3anr2  1371  chfacfscmulcl  20481  chfacfscmulgsum  20484  chfacfpmmulcl  20485  chfacfpmmulgsum  20488  cpmadumatpolylem1  20505  cpmadumatpolylem2  20506  cpmadumatpoly  20507  chcoeffeqlem  20509  2atlt  33743
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