Theorem 3bitr3d

Description: Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 24-Apr-1996)

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

Proof

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