Theorem syl3c

Description: A syllogism inference combined with contraction. (Contributed by Alan Sare, 7-Jul-2011)

	Ref	Expression
Hypotheses	syl3c.1	⊢ ( 𝜑 → 𝜓 )
	syl3c.2	⊢ ( 𝜑 → 𝜒 )
	syl3c.3	⊢ ( 𝜑 → 𝜃 )
	syl3c.4	⊢ ( 𝜓 → ( 𝜒 → ( 𝜃 → 𝜏 ) ) )
Assertion	syl3c	⊢ ( 𝜑 → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		syl3c.1	⊢ ( 𝜑 → 𝜓 )
2		syl3c.2	⊢ ( 𝜑 → 𝜒 )
3		syl3c.3	⊢ ( 𝜑 → 𝜃 )
4		syl3c.4	⊢ ( 𝜓 → ( 𝜒 → ( 𝜃 → 𝜏 ) ) )
5	1 2 4	sylc	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
6	3 5	mpd	⊢ ( 𝜑 → 𝜏 )