Theorem syl31anc

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	$$⊢\phi \to \theta$$
	syl3Xanc.4	$$⊢\phi \to \tau$$
	syl31anc.5	$$⊢\left(\left(\psi \wedge \chi \wedge \theta \right)\wedge \tau \right)\to \eta$$
Assertion	syl31anc	$$⊢\phi \to \eta$$

Proof

Step	Hyp	Ref	Expression
1		syl3anc.1	$$⊢\phi \to \psi$$
2		syl3anc.2	$$⊢\phi \to \chi$$
3		syl3anc.3	$$⊢\phi \to \theta$$
4		syl3Xanc.4	$$⊢\phi \to \tau$$
5		syl31anc.5	$$⊢\left(\left(\psi \wedge \chi \wedge \theta \right)\wedge \tau \right)\to \eta$$
6	1 2 3	3jca	$$⊢\phi \to \left(\psi \wedge \chi \wedge \theta \right)$$
7	6 4 5	syl2anc	$$⊢\phi \to \eta$$