Theorem bitrid

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

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

Proof

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