Theorem sylancb

Description: A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004)

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

Proof

Step	Hyp	Ref	Expression
1		sylancb.1	$$⊢\phi \leftrightarrow \psi$$
2		sylancb.2	$$⊢\phi \leftrightarrow \chi$$
3		sylancb.3	$$⊢\left(\psi \wedge \chi \right)\to \theta$$
4	1 2 3	syl2anb	$$⊢\left(\phi \wedge \phi \right)\to \theta$$
5	4	anidms	$$⊢\phi \to \theta$$