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Theorem syl3anl3 1523
Description: A syllogism inference. (Contributed by NM, 24-Feb-2005.)
Hypotheses
Ref Expression
syl3anl3.1 (𝜑𝜃)
syl3anl3.2 (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)
Assertion
Ref Expression
syl3anl3 (((𝜓𝜒𝜑) ∧ 𝜏) → 𝜂)

Proof of Theorem syl3anl3
StepHypRef Expression
1 syl3anl3.1 . . 3 (𝜑𝜃)
213anim3i 1157 . 2 ((𝜓𝜒𝜑) → (𝜓𝜒𝜃))
3 syl3anl3.2 . 2 (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)
42, 3sylan 569 1 (((𝜓𝜒𝜑) ∧ 𝜏) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071
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 383  df-3an 1073
This theorem is referenced by:  lgsdirnn0  25290  rdgeqoa  33555  atcvreq0  35123  paddasslem16  35644
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