Theorem nfxp

Description: Bound-variable hypothesis builder for Cartesian product. (Contributed by NM, 15-Sep-2003) (Revised by Mario Carneiro, 15-Oct-2016)

	Ref	Expression
Hypotheses	nfxp.1	|- F/_ x A
	nfxp.2	|- F/_ x B
Assertion	nfxp	|- F/_ x ( A X. B )

Proof

Step	Hyp	Ref	Expression
1		nfxp.1	|- F/_ x A
2		nfxp.2	|- F/_ x B
3		df-xp	|- ( A X. B ) = { <. y , z >. | ( y e. A /\ z e. B ) }
4	1	nfcri	|- F/ x y e. A
5	2	nfcri	|- F/ x z e. B
6	4 5	nfan	|- F/ x ( y e. A /\ z e. B )
7	6	nfopab	|- F/_ x { <. y , z >. | ( y e. A /\ z e. B ) }
8	3 7	nfcxfr	|- F/_ x ( A X. B )