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Metamath Proof Explorer

Theorem 3bitr4i

Description: A chained inference from transitive law for logical equivalence. This inference is frequently used to apply a definition to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993)

	Ref	Expression
Hypotheses	3bitr4i.1	⊢ ( 𝜑 ↔ 𝜓 )
	3bitr4i.2	⊢ ( 𝜒 ↔ 𝜑 )
	3bitr4i.3	⊢ ( 𝜃 ↔ 𝜓 )
Assertion	3bitr4i	⊢ ( 𝜒 ↔ 𝜃 )

Proof

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