Theorem sylancl

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

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

Proof

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