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Theorem sylanl1 394
Description: A syllogism inference. (Contributed by NM, 10-Mar-2005.)
Hypotheses
Ref Expression
sylanl1.1 (𝜑𝜓)
sylanl1.2 (((𝜓𝜒) ∧ 𝜃) → 𝜏)
Assertion
Ref Expression
sylanl1 (((𝜑𝜒) ∧ 𝜃) → 𝜏)

Proof of Theorem sylanl1
StepHypRef Expression
1 sylanl1.1 . . 3 (𝜑𝜓)
21anim1i 333 . 2 ((𝜑𝜒) → (𝜓𝜒))
3 sylanl1.2 . 2 (((𝜓𝜒) ∧ 𝜃) → 𝜏)
42, 3sylan 277 1 (((𝜑𝜒) ∧ 𝜃) → 𝜏)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem is referenced by:  adantlll  464  adantllr  465  isocnv  5551  mapxpen  6516  nqnq0pi  6941  nqpnq0nq  6956  addnqprl  7032  addnqpru  7033
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