Theorem syl312anc

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

	Ref	Expression
Hypotheses	syl3anc.1	$$⊢\phi \to \psi$$
	syl3anc.2	$$⊢\phi \to \chi$$
	syl3anc.3	$$⊢\phi \to \theta$$
	syl3Xanc.4	$$⊢\phi \to \tau$$
	syl23anc.5	$$⊢\phi \to \eta$$
	syl33anc.6	$$⊢\phi \to \zeta$$
	syl312anc.7	$$⊢\left(\left(\psi \wedge \chi \wedge \theta \right)\wedge \tau \wedge \left(\eta \wedge \zeta \right)\right)\to \sigma$$
Assertion	syl312anc	$$⊢\phi \to \sigma$$

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		syl23anc.5	$$⊢\phi \to \eta$$
6		syl33anc.6	$$⊢\phi \to \zeta$$
7		syl312anc.7	$$⊢\left(\left(\psi \wedge \chi \wedge \theta \right)\wedge \tau \wedge \left(\eta \wedge \zeta \right)\right)\to \sigma$$
8	5 6	jca	$$⊢\phi \to \left(\eta \wedge \zeta \right)$$
9	1 2 3 4 8 7	syl311anc	$$⊢\phi \to \sigma$$