Theorem bitr3di

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

	Ref	Expression
Hypotheses	bitr3di.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	bitr3di.2	⊢ ( 𝜓 ↔ 𝜃 )
Assertion	bitr3di	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		bitr3di.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		bitr3di.2	⊢ ( 𝜓 ↔ 𝜃 )
3	2	bicomi	⊢ ( 𝜃 ↔ 𝜓 )
4	3 1	syl5rbb	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜃 ) )