Theorem syl21anc

Description: Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009)

	Ref	Expression
Hypotheses	syl12anc.1	$$⊢\phi \to \psi$$
	syl12anc.2	$$⊢\phi \to \chi$$
	syl12anc.3	$$⊢\phi \to \theta$$
	syl21anc.4	$$⊢\left(\left(\psi \wedge \chi \right)\wedge \theta \right)\to \tau$$
Assertion	syl21anc	$$⊢\phi \to \tau$$

Proof

Step	Hyp	Ref	Expression
1		syl12anc.1	$$⊢\phi \to \psi$$
2		syl12anc.2	$$⊢\phi \to \chi$$
3		syl12anc.3	$$⊢\phi \to \theta$$
4		syl21anc.4	$$⊢\left(\left(\psi \wedge \chi \right)\wedge \theta \right)\to \tau$$
5	1 2	jca	$$⊢\phi \to \left(\psi \wedge \chi \right)$$
6	5 3 4	syl2anc	$$⊢\phi \to \tau$$