Theorem bitrid

Description: A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993)

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

Proof

Step	Hyp	Ref	Expression
1		bitrid.1	$$⊢\phi \leftrightarrow \psi$$
2		bitrid.2	$$⊢\chi \to \left(\psi \leftrightarrow \theta \right)$$
3	1	a1i	$$⊢\chi \to \left(\phi \leftrightarrow \psi \right)$$
4	3 2	bitrd	$$⊢\chi \to \left(\phi \leftrightarrow \theta \right)$$