Theorem breq12d

Description: Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996) (Proof shortened by Andrew Salmon, 9-Jul-2011)

	Ref	Expression
Hypotheses	breq1d.1	$$⊢\phi \to A=B$$
	breq12d.2	$$⊢\phi \to C=D$$
Assertion	breq12d	$$⊢\phi \to \left(ARC\leftrightarrow BRD\right)$$

Proof

Step	Hyp	Ref	Expression
1		breq1d.1	$$⊢\phi \to A=B$$
2		breq12d.2	$$⊢\phi \to C=D$$
3		breq12	$$⊢\left(A=B\wedge C=D\right)\to \left(ARC\leftrightarrow BRD\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(ARC\leftrightarrow BRD\right)$$