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Theorem syl3an12 40103
Description: A double syllogism inference. (Contributed by SN, 15-Sep-2024.)
Hypotheses
Ref Expression
syl3an12.1 (𝜑𝜓)
syl3an12.2 (𝜒𝜃)
syl3an12.s ((𝜓𝜃𝜏) → 𝜂)
Assertion
Ref Expression
syl3an12 ((𝜑𝜒𝜏) → 𝜂)

Proof of Theorem syl3an12
StepHypRef Expression
1 syl3an12.1 . 2 (𝜑𝜓)
2 syl3an12.2 . 2 (𝜒𝜃)
3 id 22 . 2 (𝜏𝜏)
4 syl3an12.s . 2 ((𝜓𝜃𝜏) → 𝜂)
51, 2, 3, 4syl3an 1158 1 ((𝜑𝜒𝜏) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1085
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 206  df-an 396  df-3an 1087
This theorem is referenced by:  dvdsexpb  40263
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