Theorem nfxp 4668

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

Hypotheses

Ref	Expression
nfxp.1	|- F/_ x A
nfxp.2	|- F/_ x B

Assertion

Ref	Expression
nfxp	|- F/_ x ( A X. B )

Proof of Theorem nfxp

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-xp 4647	. 2 |- ( A X. B ) = { <. y , z >. | ( y e. A /\ z e. B ) }
2		nfxp.1	. . . . 5 |- F/_ x A
3	2	nfcri 2326	. . . 4 |- F/ x y e. A
4		nfxp.2	. . . . 5 |- F/_ x B
5	4	nfcri 2326	. . . 4 |- F/ x z e. B
6	3, 5	nfan 1576	. . 3 |- F/ x ( y e. A /\ z e. B )
7	6	nfopab 4086	. 2 |- F/_ x { <. y , z >. | ( y e. A /\ z e. B ) }
8	1, 7	nfcxfr 2329	1 |- F/_ x ( A X. B )

Syntax hints:   /\ wa 104   e. wcel 2160   F/_wnfc 2319   {copab 4078   X. cxp 4639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-opab 4080  df-xp 4647

This theorem is referenced by:  opeliunxp  4696  nfres  4924  mpomptsx  6217  dmmpossx  6219  fmpox  6220  disjxp1  6256  nfdju  7066  fsum2dlemstep  11469  fisumcom2  11473  fprod2dlemstep  11657  fprodcom2fi  11661