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Theorem syl3anr2 1199
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 259 . . 3 (((𝜓𝜃𝜏) ∧ 𝜒) → 𝜂)
41, 3syl3anl2 1195 . 2 (((𝜓𝜑𝜏) ∧ 𝜒) → 𝜂)
54ancoms 259 1 ((𝜒 ∧ (𝜓𝜑𝜏)) → 𝜂)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3a 896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105
This theorem depends on definitions:  df-bi 114  df-3an 898
This theorem is referenced by: (None)
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