Metamath Proof Explorer


Theorem syl3an12

Description: A double syllogism inference. (Contributed by SN, 15-Sep-2024)

Ref Expression
Hypotheses syl3an12.1 ( 𝜑𝜓 )
syl3an12.2 ( 𝜒𝜃 )
syl3an12.s ( ( 𝜓𝜃𝜏 ) → 𝜂 )
Assertion syl3an12 ( ( 𝜑𝜒𝜏 ) → 𝜂 )

Proof

Step Hyp Ref Expression
1 syl3an12.1 ( 𝜑𝜓 )
2 syl3an12.2 ( 𝜒𝜃 )
3 syl3an12.s ( ( 𝜓𝜃𝜏 ) → 𝜂 )
4 id ( 𝜏𝜏 )
5 1 2 4 3 syl3an ( ( 𝜑𝜒𝜏 ) → 𝜂 )