Theorem syl232anc

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	$$⊢\phi \to \zeta$$
	syl133anc.7	$$⊢\phi \to \sigma$$
	syl232anc.8	$$⊢\left(\left(\psi \wedge \chi \right)\wedge \left(\theta \wedge \tau \wedge \eta \right)\wedge \left(\zeta \wedge \sigma \right)\right)\to \rho$$
Assertion	syl232anc	$$⊢\phi \to \rho$$

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		syl133anc.7	$$⊢\phi \to \sigma$$
8		syl232anc.8	$$⊢\left(\left(\psi \wedge \chi \right)\wedge \left(\theta \wedge \tau \wedge \eta \right)\wedge \left(\zeta \wedge \sigma \right)\right)\to \rho$$
9	6 7	jca	$$⊢\phi \to \left(\zeta \wedge \sigma \right)$$
10	1 2 3 4 5 9 8	syl231anc	$$⊢\phi \to \rho$$