Theorem 3bitr3rd

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

	Ref	Expression
Hypotheses	3bitr3d.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	3bitr3d.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )
	3bitr3d.3	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜏 ) )
Assertion	3bitr3rd	⊢ ( 𝜑 → ( 𝜏 ↔ 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		3bitr3d.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		3bitr3d.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )
3		3bitr3d.3	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜏 ) )
4	1 2	bitr3d	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜃 ) )
5	3 4	bitr3d	⊢ ( 𝜑 → ( 𝜏 ↔ 𝜃 ) )