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Theorem sylanl2 400
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 339 . 2 ((𝜓𝜑) → (𝜓𝜒))
3 sylanl2.2 . 2 (((𝜓𝜒) ∧ 𝜃) → 𝜏)
42, 3sylan 281 1 (((𝜓𝜑) ∧ 𝜃) → 𝜏)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem is referenced by:  mpanlr1  436  adantlrl  473  adantlrr  474  cnegexlem3  7907  mulsub  8131  divsubdivap  8456  modqcyc2  10101  lcmneg  11682
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