Theorem syl221anc

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	$$⊢\phi \to \tau$$
	syl23anc.5	$$⊢\phi \to \eta$$
	syl221anc.6	$$⊢\left(\left(\psi \wedge \chi \right)\wedge \left(\theta \wedge \tau \right)\wedge \eta \right)\to \zeta$$
Assertion	syl221anc	$$⊢\phi \to \zeta$$

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		syl221anc.6	$$⊢\left(\left(\psi \wedge \chi \right)\wedge \left(\theta \wedge \tau \right)\wedge \eta \right)\to \zeta$$
7	3 4	jca	$$⊢\phi \to \left(\theta \wedge \tau \right)$$
8	1 2 7 5 6	syl211anc	$$⊢\phi \to \zeta$$