Theorem eqbrtrd

Description: Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999)

	Ref	Expression
Hypotheses	eqbrtrd.1	$$⊢\phi \to A=B$$
	eqbrtrd.2	$$⊢\phi \to BRC$$
Assertion	eqbrtrd	$$⊢\phi \to ARC$$

Proof

Step	Hyp	Ref	Expression
1		eqbrtrd.1	$$⊢\phi \to A=B$$
2		eqbrtrd.2	$$⊢\phi \to BRC$$
3	1	breq1d	$$⊢\phi \to \left(ARC\leftrightarrow BRC\right)$$
4	2 3	mpbird	$$⊢\phi \to ARC$$