Theorem sylancbr

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

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

Proof

Step	Hyp	Ref	Expression
1		sylancbr.1	$$⊢\psi \leftrightarrow \phi$$
2		sylancbr.2	$$⊢\chi \leftrightarrow \phi$$
3		sylancbr.3	$$⊢\left(\psi \wedge \chi \right)\to \theta$$
4	1 2 3	syl2anbr	$$⊢\left(\phi \wedge \phi \right)\to \theta$$
5	4	anidms	$$⊢\phi \to \theta$$