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Theorem syl3anl 1369
 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 1240 . 2 ((𝜑𝜒𝜏) → (𝜓𝜃𝜂))
5 syl3anl.4 . 2 (((𝜓𝜃𝜂) ∧ 𝜁) → 𝜎)
64, 5sylan 487 1 (((𝜑𝜒𝜏) ∧ 𝜁) → 𝜎)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031 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 196  df-an 385  df-3an 1033 This theorem is referenced by:  chlej1  27753  chlej2  27754  atcvatlem  28628
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