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Theorem syl3anr2 1419
Description: A syllogism inference. (Contributed by NM, 1-Aug-2007.)
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)
2 syl3anr2.2 . . . 4 ((𝜒 ∧ (𝜓𝜃𝜏)) → 𝜂)
32ancoms 468 . . 3 (((𝜓𝜃𝜏) ∧ 𝜒) → 𝜂)
41, 3syl3anl2 1415 . 2 (((𝜓𝜑𝜏) ∧ 𝜒) → 𝜂)
54ancoms 468 1 ((𝜒 ∧ (𝜓𝜑𝜏)) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054
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 385  df-3an 1056
This theorem is referenced by:  mulgsubdir  17629  dipassr2  27830
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