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Theorem syl3anl 1268
Description: A triple syllogism inference. (Contributed by NM, 24-Dec-2006.)
Hypotheses
Ref Expression
syl3anl.1 (𝜑𝜓)
syl3anl.2 (𝜒𝜃)
syl3anl.3 (𝜏𝜂)
syl3anl.4 (((𝜓𝜃𝜂) ∧ 𝜁) → 𝜎)
Assertion
Ref Expression
syl3anl (((𝜑𝜒𝜏) ∧ 𝜁) → 𝜎)

Proof of Theorem syl3anl
StepHypRef Expression
1 syl3anl.1 . . 3 (𝜑𝜓)
2 syl3anl.2 . . 3 (𝜒𝜃)
3 syl3anl.3 . . 3 (𝜏𝜂)
41, 2, 33anim123i 1167 . 2 ((𝜑𝜒𝜏) → (𝜓𝜃𝜂))
5 syl3anl.4 . 2 (((𝜓𝜃𝜂) ∧ 𝜁) → 𝜎)
64, 5sylan 281 1 (((𝜑𝜒𝜏) ∧ 𝜁) → 𝜎)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116  df-3an 965
This theorem is referenced by: (None)
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