Theorem syl2anc

Description: Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012)

	Ref	Expression
Hypotheses	syl2anc.1	$$⊢\phi \to \psi$$
	syl2anc.2	$$⊢\phi \to \chi$$
	syl2anc.3	$$⊢\left(\psi \wedge \chi \right)\to \theta$$
Assertion	syl2anc	$$⊢\phi \to \theta$$

Proof

Step	Hyp	Ref	Expression
1		syl2anc.1	$$⊢\phi \to \psi$$
2		syl2anc.2	$$⊢\phi \to \chi$$
3		syl2anc.3	$$⊢\left(\psi \wedge \chi \right)\to \theta$$
4	3	ex	$$⊢\psi \to \left(\chi \to \theta \right)$$
5	1 2 4	sylc	$$⊢\phi \to \theta$$