Theorem syl5bir

Description: A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Hypotheses	syl5bir.1	$$⊢\psi \leftrightarrow \phi$$
	syl5bir.2	$$⊢\chi \to \left(\psi \to \theta \right)$$
Assertion	syl5bir	$$⊢\chi \to \left(\phi \to \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		syl5bir.1	$$⊢\psi \leftrightarrow \phi$$
2		syl5bir.2	$$⊢\chi \to \left(\psi \to \theta \right)$$
3	1	biimpri	$$⊢\phi \to \psi$$
4	3 2	syl5	$$⊢\chi \to \left(\phi \to \theta \right)$$