Theorem xrneq12d

Description: Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 18-Dec-2021)

	Ref	Expression
Hypotheses	xrneq12d.1	$$⊢\phi \to A=B$$
	xrneq12d.2	$$⊢\phi \to C=D$$
Assertion	xrneq12d	$$⊢\phi \to A⋉C=B⋉D$$

Proof

Step	Hyp	Ref	Expression
1		xrneq12d.1	$$⊢\phi \to A=B$$
2		xrneq12d.2	$$⊢\phi \to C=D$$
3		xrneq12	$$⊢\left(A=B\wedge C=D\right)\to A⋉C=B⋉D$$
4	1 2 3	syl2anc	$$⊢\phi \to A⋉C=B⋉D$$