Theorem breq1

Description: Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993)

	Ref	Expression
Assertion	breq1	$$⊢A=B\to \left(ARC\leftrightarrow BRC\right)$$

Proof

Step	Hyp	Ref	Expression
1		opeq1	$$⊢A=B\to ⟨A,C⟩=⟨B,C⟩$$
2	1	eleq1d	$$⊢A=B\to \left(⟨A,C⟩\in R\leftrightarrow ⟨B,C⟩\in R\right)$$
3		df-br	$$⊢ARC\leftrightarrow ⟨A,C⟩\in R$$
4		df-br	$$⊢BRC\leftrightarrow ⟨B,C⟩\in R$$
5	2 3 4	3bitr4g	$$⊢A=B\to \left(ARC\leftrightarrow BRC\right)$$