Theorem syl6ibr

Description: A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded consequent with a definition. (Contributed by NM, 10-Jan-1993)

	Ref	Expression
Hypotheses	syl6ibr.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
	syl6ibr.2	$$⊢\theta \leftrightarrow \chi$$
Assertion	syl6ibr	$$⊢\phi \to \left(\psi \to \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		syl6ibr.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		syl6ibr.2	$$⊢\theta \leftrightarrow \chi$$
3	2	biimpri	$$⊢\chi \to \theta$$
4	1 3	syl6	$$⊢\phi \to \left(\psi \to \theta \right)$$