Theorem syl213anc

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	$$⊢\phi \to \eta$$
	syl33anc.6	$$⊢\phi \to \zeta$$
	syl213anc.7	$$⊢\left(\left(\psi \wedge \chi \right)\wedge \theta \wedge \left(\tau \wedge \eta \wedge \zeta \right)\right)\to \sigma$$
Assertion	syl213anc	$$⊢\phi \to \sigma$$

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		syl213anc.7	$$⊢\left(\left(\psi \wedge \chi \right)\wedge \theta \wedge \left(\tau \wedge \eta \wedge \zeta \right)\right)\to \sigma$$
8	1 2	jca	$$⊢\phi \to \left(\psi \wedge \chi \right)$$
9	8 3 4 5 6 7	syl113anc	$$⊢\phi \to \sigma$$