Theorem bitr4id

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

	Ref	Expression
Hypotheses	bitr4id.2	$$⊢\psi \leftrightarrow \chi$$
	bitr4id.1	$$⊢\phi \to \left(\theta \leftrightarrow \chi \right)$$
Assertion	bitr4id	$$⊢\phi \to \left(\psi \leftrightarrow \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		bitr4id.2	$$⊢\psi \leftrightarrow \chi$$
2		bitr4id.1	$$⊢\phi \to \left(\theta \leftrightarrow \chi \right)$$
3	1	bicomi	$$⊢\chi \leftrightarrow \psi$$
4	2 3	bitr2di	$$⊢\phi \to \left(\psi \leftrightarrow \theta \right)$$