Theorem 3bitr4rd

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

	Ref	Expression
Hypotheses	3bitr4d.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	3bitr4d.2	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜓 ) )
	3bitr4d.3	⊢ ( 𝜑 → ( 𝜏 ↔ 𝜒 ) )
Assertion	3bitr4rd	⊢ ( 𝜑 → ( 𝜏 ↔ 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		3bitr4d.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		3bitr4d.2	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜓 ) )
3		3bitr4d.3	⊢ ( 𝜑 → ( 𝜏 ↔ 𝜒 ) )
4	3 1	bitr4d	⊢ ( 𝜑 → ( 𝜏 ↔ 𝜓 ) )
5	4 2	bitr4d	⊢ ( 𝜑 → ( 𝜏 ↔ 𝜃 ) )