Theorem syl5rbb

Description: A syllogism inference from two biconditionals. (Contributed by NM, 1-Aug-1993)

	Ref	Expression
Hypotheses	syl5rbb.1	⊢ ( 𝜑 ↔ 𝜓 )
	syl5rbb.2	⊢ ( 𝜒 → ( 𝜓 ↔ 𝜃 ) )
Assertion	syl5rbb	⊢ ( 𝜒 → ( 𝜃 ↔ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		syl5rbb.1	⊢ ( 𝜑 ↔ 𝜓 )
2		syl5rbb.2	⊢ ( 𝜒 → ( 𝜓 ↔ 𝜃 ) )
3	1 2	syl5bb	⊢ ( 𝜒 → ( 𝜑 ↔ 𝜃 ) )
4	3	bicomd	⊢ ( 𝜒 → ( 𝜃 ↔ 𝜑 ) )