Theorem 3bitrri

Description: A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006)

	Ref	Expression
Hypotheses	3bitri.1	⊢ ( 𝜑 ↔ 𝜓 )
	3bitri.2	⊢ ( 𝜓 ↔ 𝜒 )
	3bitri.3	⊢ ( 𝜒 ↔ 𝜃 )
Assertion	3bitrri	⊢ ( 𝜃 ↔ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		3bitri.1	⊢ ( 𝜑 ↔ 𝜓 )
2		3bitri.2	⊢ ( 𝜓 ↔ 𝜒 )
3		3bitri.3	⊢ ( 𝜒 ↔ 𝜃 )
4	1 2	bitr2i	⊢ ( 𝜒 ↔ 𝜑 )
5	3 4	bitr3i	⊢ ( 𝜃 ↔ 𝜑 )