Theorem xrneq12d

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

	Ref	Expression
Hypotheses	xrneq12d.1	|- ( ph -> A = B )
	xrneq12d.2	|- ( ph -> C = D )
Assertion	xrneq12d	|- ( ph -> ( A |X. C ) = ( B |X. D ) )

Proof

Step	Hyp	Ref	Expression
1		xrneq12d.1	|- ( ph -> A = B )
2		xrneq12d.2	|- ( ph -> C = D )
3		xrneq12	|- ( ( A = B /\ C = D ) -> ( A |X. C ) = ( B |X. D ) )
4	1 2 3	syl2anc	|- ( ph -> ( A |X. C ) = ( B |X. D ) )