Theorem biimtrid

Description: A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded antecedent with a definition. (Contributed by NM, 12-Jan-1993)

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

Proof

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