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Theorem sylanl2 383
Description: A syllogism inference. (Contributed by NM, 1-Jan-2005.)
Hypotheses
Ref Expression
sylanl2.1 (𝜑𝜒)
sylanl2.2 (((𝜓𝜒) ∧ 𝜃) → 𝜏)
Assertion
Ref Expression
sylanl2 (((𝜓𝜑) ∧ 𝜃) → 𝜏)

Proof of Theorem sylanl2
StepHypRef Expression
1 sylanl2.1 . . 3 (𝜑𝜒)
21anim2i 324 . 2 ((𝜓𝜑) → (𝜓𝜒))
3 sylanl2.2 . 2 (((𝜓𝜒) ∧ 𝜃) → 𝜏)
42, 3sylan 267 1 (((𝜓𝜑) ∧ 𝜃) → 𝜏)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101
This theorem is referenced by:  mpanlr1  416  adantlrl  451  adantlrr  452  cnegexlem3  7186  mulsub  7396  divsubdivap  7702
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