Theorem nfxp 4631

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 4610	. 2 |- ( A X. B ) = { <. y , z >. | ( y e. A /\ z e. B ) }
2		nfxp.1	. . . . 5 |- F/_ x A
3	2	nfcri 2302	. . . 4 |- F/ x y e. A
4		nfxp.2	. . . . 5 |- F/_ x B
5	4	nfcri 2302	. . . 4 |- F/ x z e. B
6	3, 5	nfan 1553	. . 3 |- F/ x ( y e. A /\ z e. B )
7	6	nfopab 4050	. 2 |- F/_ x { <. y , z >. | ( y e. A /\ z e. B ) }
8	1, 7	nfcxfr 2305	1 |- F/_ x ( A X. B )

Syntax hints:   /\ wa 103   e. wcel 2136   F/_wnfc 2295   {copab 4042   X. cxp 4602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-opab 4044  df-xp 4610

This theorem is referenced by:  opeliunxp  4659  nfres  4886  mpomptsx  6165  dmmpossx  6167  fmpox  6168  disjxp1  6204  nfdju  7007  fsum2dlemstep  11375  fisumcom2  11379  fprod2dlemstep  11563  fprodcom2fi  11567