Theorem eqbrtrdi

Description: A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999)

	Ref	Expression
Hypotheses	eqbrtrdi.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	eqbrtrdi.2	⊢ 𝐵 𝑅 𝐶
Assertion	eqbrtrdi	⊢ ( 𝜑 → 𝐴 𝑅 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eqbrtrdi.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		eqbrtrdi.2	⊢ 𝐵 𝑅 𝐶
3	1	breq1d	⊢ ( 𝜑 → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐶 ) )
4	2 3	mpbiri	⊢ ( 𝜑 → 𝐴 𝑅 𝐶 )