Theorem bitr3id

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

	Ref	Expression
Hypotheses	bitr3id.1	⊢ ( 𝜓 ↔ 𝜑 )
	bitr3id.2	⊢ ( 𝜒 → ( 𝜓 ↔ 𝜃 ) )
Assertion	bitr3id	⊢ ( 𝜒 → ( 𝜑 ↔ 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		bitr3id.1	⊢ ( 𝜓 ↔ 𝜑 )
2		bitr3id.2	⊢ ( 𝜒 → ( 𝜓 ↔ 𝜃 ) )
3	1	bicomi	⊢ ( 𝜑 ↔ 𝜓 )
4	3 2	syl5bb	⊢ ( 𝜒 → ( 𝜑 ↔ 𝜃 ) )