Metamath Proof Explorer


Theorem syl6ci

Description: A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012)

Ref Expression
Hypotheses syl6ci.1 ( 𝜑 → ( 𝜓𝜒 ) )
syl6ci.2 ( 𝜑𝜃 )
syl6ci.3 ( 𝜒 → ( 𝜃𝜏 ) )
Assertion syl6ci ( 𝜑 → ( 𝜓𝜏 ) )

Proof

Step Hyp Ref Expression
1 syl6ci.1 ( 𝜑 → ( 𝜓𝜒 ) )
2 syl6ci.2 ( 𝜑𝜃 )
3 syl6ci.3 ( 𝜒 → ( 𝜃𝜏 ) )
4 2 a1d ( 𝜑 → ( 𝜓𝜃 ) )
5 1 4 3 syl6c ( 𝜑 → ( 𝜓𝜏 ) )