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Theorem syl3anr1 1375
Description: A syllogism inference. (Contributed by NM, 31-Jul-2007.)
Hypotheses
Ref Expression
syl3anr1.1 (𝜑𝜓)
syl3anr1.2 ((𝜒 ∧ (𝜓𝜃𝜏)) → 𝜂)
Assertion
Ref Expression
syl3anr1 ((𝜒 ∧ (𝜑𝜃𝜏)) → 𝜂)

Proof of Theorem syl3anr1
StepHypRef Expression
1 syl3anr1.1 . . 3 (𝜑𝜓)
213anim1i 1246 . 2 ((𝜑𝜃𝜏) → (𝜓𝜃𝜏))
3 syl3anr1.2 . 2 ((𝜒 ∧ (𝜓𝜃𝜏)) → 𝜂)
42, 3sylan2 491 1 ((𝜒 ∧ (𝜑𝜃𝜏)) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036
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 386  df-3an 1038
This theorem is referenced by:  btwnconn1lem4  31860  pridlc2  33524  atmod1i1  34644  prmdvdsfmtnof1lem2  40812
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