Theorem bitrdi

Description: A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993)

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

Proof

Step	Hyp	Ref	Expression
1		bitrdi.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		bitrdi.2	⊢ ( 𝜒 ↔ 𝜃 )
3	2	a1i	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜃 ) )
4	1 3	bitrd	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )