Theorem sylancr

Description: Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009)

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

Proof

Step	Hyp	Ref	Expression
1		sylancr.1	$$⊢\psi$$
2		sylancr.2	$$⊢\phi \to \chi$$
3		sylancr.3	$$⊢\left(\psi \wedge \chi \right)\to \theta$$
4	1	a1i	$$⊢\phi \to \psi$$
5	4 2 3	syl2anc	$$⊢\phi \to \theta$$