Theorem bitr3di

Description: A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994)

	Ref	Expression
Hypotheses	bitr3di.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	bitr3di.2	$$⊢\psi \leftrightarrow \theta$$
Assertion	bitr3di	$$⊢\phi \to \left(\chi \leftrightarrow \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		bitr3di.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		bitr3di.2	$$⊢\psi \leftrightarrow \theta$$
3	2	bicomi	$$⊢\theta \leftrightarrow \psi$$
4	3 1	bitr2id	$$⊢\phi \to \left(\chi \leftrightarrow \theta \right)$$