Metamath Proof Explorer


Theorem syl3an3

Description: A syllogism inference. (Contributed by NM, 22-Aug-1995) (Proof shortened by Wolf Lammen, 26-Jun-2022)

Ref Expression
Hypotheses syl3an3.1 ( 𝜑𝜃 )
syl3an3.2 ( ( 𝜓𝜒𝜃 ) → 𝜏 )
Assertion syl3an3 ( ( 𝜓𝜒𝜑 ) → 𝜏 )

Proof

Step Hyp Ref Expression
1 syl3an3.1 ( 𝜑𝜃 )
2 syl3an3.2 ( ( 𝜓𝜒𝜃 ) → 𝜏 )
3 1 3anim3i ( ( 𝜓𝜒𝜑 ) → ( 𝜓𝜒𝜃 ) )
4 3 2 syl ( ( 𝜓𝜒𝜑 ) → 𝜏 )