Theorem syl22anc

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

	Ref	Expression
Hypotheses	syl12anc.1	$$⊢\phi \to \psi$$
	syl12anc.2	$$⊢\phi \to \chi$$
	syl12anc.3	$$⊢\phi \to \theta$$
	syl22anc.4	$$⊢\phi \to \tau$$
	syl22anc.5	$$⊢\left(\left(\psi \wedge \chi \right)\wedge \left(\theta \wedge \tau \right)\right)\to \eta$$
Assertion	syl22anc	$$⊢\phi \to \eta$$

Proof

Step	Hyp	Ref	Expression
1		syl12anc.1	$$⊢\phi \to \psi$$
2		syl12anc.2	$$⊢\phi \to \chi$$
3		syl12anc.3	$$⊢\phi \to \theta$$
4		syl22anc.4	$$⊢\phi \to \tau$$
5		syl22anc.5	$$⊢\left(\left(\psi \wedge \chi \right)\wedge \left(\theta \wedge \tau \right)\right)\to \eta$$
6	1 2	jca	$$⊢\phi \to \left(\psi \wedge \chi \right)$$
7	6 3 4 5	syl12anc	$$⊢\phi \to \eta$$