Theorem breqtrd

Description: Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999)

	Ref	Expression
Hypotheses	breqtrd.1	$$⊢\phi \to ARB$$
	breqtrd.2	$$⊢\phi \to B=C$$
Assertion	breqtrd	$$⊢\phi \to ARC$$

Proof

Step	Hyp	Ref	Expression
1		breqtrd.1	$$⊢\phi \to ARB$$
2		breqtrd.2	$$⊢\phi \to B=C$$
3	2	breq2d	$$⊢\phi \to \left(ARB\leftrightarrow ARC\right)$$
4	1 3	mpbid	$$⊢\phi \to ARC$$