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Theorem syl3anr2 1414
Description: A syllogism inference. (Contributed by NM, 1-Aug-2007.) (Proof shortened by Wolf Lammen, 27-Jun-2022.)
Hypotheses
Ref Expression
syl3anr2.1 (𝜑𝜃)
syl3anr2.2 ((𝜒 ∧ (𝜓𝜃𝜏)) → 𝜂)
Assertion
Ref Expression
syl3anr2 ((𝜒 ∧ (𝜓𝜑𝜏)) → 𝜂)

Proof of Theorem syl3anr2
StepHypRef Expression
1 syl3anr2.1 . . 3 (𝜑𝜃)
213anim2i 1150 . 2 ((𝜓𝜑𝜏) → (𝜓𝜃𝜏))
3 syl3anr2.2 . 2 ((𝜒 ∧ (𝜓𝜃𝜏)) → 𝜂)
42, 3sylan2 595 1 ((𝜒 ∧ (𝜓𝜑𝜏)) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084
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 210  df-an 400  df-3an 1086
This theorem is referenced by:  mulgsubdir  18262  dipassr2  28633
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