Theorem 3bitr2i

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

	Ref	Expression
Hypotheses	3bitr2i.1	⊢ ( 𝜑 ↔ 𝜓 )
	3bitr2i.2	⊢ ( 𝜒 ↔ 𝜓 )
	3bitr2i.3	⊢ ( 𝜒 ↔ 𝜃 )
Assertion	3bitr2i	⊢ ( 𝜑 ↔ 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		3bitr2i.1	⊢ ( 𝜑 ↔ 𝜓 )
2		3bitr2i.2	⊢ ( 𝜒 ↔ 𝜓 )
3		3bitr2i.3	⊢ ( 𝜒 ↔ 𝜃 )
4	1 2	bitr4i	⊢ ( 𝜑 ↔ 𝜒 )
5	4 3	bitri	⊢ ( 𝜑 ↔ 𝜃 )