Theorem 3bitrrd

Description: Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006)

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

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	bitr2d	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜓 ) )
5	3 4	bitr3d	⊢ ( 𝜑 → ( 𝜏 ↔ 𝜓 ) )