Theorem bitr2di

Description: A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993)

	Ref	Expression
Hypotheses	bitr2di.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	bitr2di.2	⊢ ( 𝜒 ↔ 𝜃 )
Assertion	bitr2di	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		bitr2di.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		bitr2di.2	⊢ ( 𝜒 ↔ 𝜃 )
3	1 2	bitrdi	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )
4	3	bicomd	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜓 ) )