Theorem syl6c

Description: Inference combining syl6 with contraction. (Contributed by Alan Sare, 2-May-2011)

	Ref	Expression
Hypotheses	syl6c.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
	syl6c.2	$$⊢\phi \to \left(\psi \to \theta \right)$$
	syl6c.3	$$⊢\chi \to \left(\theta \to \tau \right)$$
Assertion	syl6c	$$⊢\phi \to \left(\psi \to \tau \right)$$

Proof

Step	Hyp	Ref	Expression
1		syl6c.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		syl6c.2	$$⊢\phi \to \left(\psi \to \theta \right)$$
3		syl6c.3	$$⊢\chi \to \left(\theta \to \tau \right)$$
4	1 3	syl6	$$⊢\phi \to \left(\psi \to \left(\theta \to \tau \right)\right)$$
5	2 4	mpdd	$$⊢\phi \to \left(\psi \to \tau \right)$$