Theorem breq123d

Description: Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011)

	Ref	Expression
Hypotheses	breq1d.1	$$⊢\phi \to A=B$$
	breq123d.2	$$⊢\phi \to R=S$$
	breq123d.3	$$⊢\phi \to C=D$$
Assertion	breq123d	$$⊢\phi \to \left(ARC\leftrightarrow BSD\right)$$

Proof

Step	Hyp	Ref	Expression
1		breq1d.1	$$⊢\phi \to A=B$$
2		breq123d.2	$$⊢\phi \to R=S$$
3		breq123d.3	$$⊢\phi \to C=D$$
4	1 3	breq12d	$$⊢\phi \to \left(ARC\leftrightarrow BRD\right)$$
5	2	breqd	$$⊢\phi \to \left(BRD\leftrightarrow BSD\right)$$
6	4 5	bitrd	$$⊢\phi \to \left(ARC\leftrightarrow BSD\right)$$