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Theorem syl3anl2 1372
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 450 . . 3 ((𝜓𝜒𝜃) → (𝜏𝜂))
41, 3syl3an2 1357 . 2 ((𝜓𝜑𝜃) → (𝜏𝜂))
54imp 445 1 (((𝜓𝜑𝜃) ∧ 𝜏) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036
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 197  df-an 386  df-3an 1038
This theorem is referenced by:  syl3anr2  1376  chfacfscmulcl  20581  chfacfscmulgsum  20584  chfacfpmmulcl  20585  chfacfpmmulgsum  20588  cpmadumatpolylem1  20605  cpmadumatpolylem2  20606  cpmadumatpoly  20607  chcoeffeqlem  20609  2atlt  34205
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