Theorem syl3c

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

	Ref	Expression
Hypotheses	syl3c.1	$$⊢\phi \to \psi$$
	syl3c.2	$$⊢\phi \to \chi$$
	syl3c.3	$$⊢\phi \to \theta$$
	syl3c.4	$$⊢\psi \to \left(\chi \to \left(\theta \to \tau \right)\right)$$
Assertion	syl3c	$$⊢\phi \to \tau$$

Proof

Step	Hyp	Ref	Expression
1		syl3c.1	$$⊢\phi \to \psi$$
2		syl3c.2	$$⊢\phi \to \chi$$
3		syl3c.3	$$⊢\phi \to \theta$$
4		syl3c.4	$$⊢\psi \to \left(\chi \to \left(\theta \to \tau \right)\right)$$
5	1 2 4	sylc	$$⊢\phi \to \left(\theta \to \tau \right)$$
6	3 5	mpd	$$⊢\phi \to \tau$$