Theorem syl6ci

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

	Ref	Expression
Hypotheses	syl6ci.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	syl6ci.2	⊢ ( 𝜑 → 𝜃 )
	syl6ci.3	⊢ ( 𝜒 → ( 𝜃 → 𝜏 ) )
Assertion	syl6ci	⊢ ( 𝜑 → ( 𝜓 → 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		syl6ci.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		syl6ci.2	⊢ ( 𝜑 → 𝜃 )
3		syl6ci.3	⊢ ( 𝜒 → ( 𝜃 → 𝜏 ) )
4	2	a1d	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
5	1 4 3	syl6c	⊢ ( 𝜑 → ( 𝜓 → 𝜏 ) )