Theorem syl3anc

Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012)

	Ref	Expression
Hypotheses	syl3anc.1	$$⊢\phi \to \psi$$
	syl3anc.2	$$⊢\phi \to \chi$$
	syl3anc.3	$$⊢\phi \to \theta$$
	syl3anc.4	$$⊢\left(\psi \wedge \chi \wedge \theta \right)\to \tau$$
Assertion	syl3anc	$$⊢\phi \to \tau$$

Proof

Step	Hyp	Ref	Expression
1		syl3anc.1	$$⊢\phi \to \psi$$
2		syl3anc.2	$$⊢\phi \to \chi$$
3		syl3anc.3	$$⊢\phi \to \theta$$
4		syl3anc.4	$$⊢\left(\psi \wedge \chi \wedge \theta \right)\to \tau$$
5	1 2 3	3jca	$$⊢\phi \to \left(\psi \wedge \chi \wedge \theta \right)$$
6	5 4	syl	$$⊢\phi \to \tau$$