Theorem 3brtr3d

Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999)

	Ref	Expression
Hypotheses	3brtr3d.1	$$⊢\phi \to ARB$$
	3brtr3d.2	$$⊢\phi \to A=C$$
	3brtr3d.3	$$⊢\phi \to B=D$$
Assertion	3brtr3d	$$⊢\phi \to CRD$$

Proof

Step	Hyp	Ref	Expression
1		3brtr3d.1	$$⊢\phi \to ARB$$
2		3brtr3d.2	$$⊢\phi \to A=C$$
3		3brtr3d.3	$$⊢\phi \to B=D$$
4	2 3	breq12d	$$⊢\phi \to \left(ARB\leftrightarrow CRD\right)$$
5	1 4	mpbid	$$⊢\phi \to CRD$$