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Theorem syl3anl1 1520
Description: A syllogism inference. (Contributed by NM, 24-Feb-2005.)
Hypotheses
Ref Expression
syl3anl1.1 (𝜑𝜓)
syl3anl1.2 (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)
Assertion
Ref Expression
syl3anl1 (((𝜑𝜒𝜃) ∧ 𝜏) → 𝜂)

Proof of Theorem syl3anl1
StepHypRef Expression
1 syl3anl1.1 . . 3 (𝜑𝜓)
213anim1i 1155 . 2 ((𝜑𝜒𝜃) → (𝜓𝜒𝜃))
3 syl3anl1.2 . 2 (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)
42, 3sylan 569 1 (((𝜑𝜒𝜃) ∧ 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:  suprzcl  11659  latjcom  17267  latmcom  17283  ring1zr  19490  lgsdinn0  25291  crngohomfo  34137  dalem53  35533
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