Theorem bitr4id

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

	Ref	Expression
Hypotheses	bitr4id.2	⊢ ( 𝜓 ↔ 𝜒 )
	bitr4id.1	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜒 ) )
Assertion	bitr4id	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		bitr4id.2	⊢ ( 𝜓 ↔ 𝜒 )
2		bitr4id.1	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜒 ) )
3	1	bicomi	⊢ ( 𝜒 ↔ 𝜓 )
4	2 3	bitr2di	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )