Theorem 3bitrd

Description: Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999)

	Ref	Expression
Hypotheses	3bitrd.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	3bitrd.2	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜃 ) )
	3bitrd.3	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )
Assertion	3bitrd	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		3bitrd.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		3bitrd.2	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜃 ) )
3		3bitrd.3	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )
4	1 2	bitrd	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )
5	4 3	bitrd	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜏 ) )