Theorem nfxp 4703

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 4682	. 2 |- ( A X. B ) = { <. y , z >. | ( y e. A /\ z e. B ) }
2		nfxp.1	. . . . 5 |- F/_ x A
3	2	nfcri 2342	. . . 4 |- F/ x y e. A
4		nfxp.2	. . . . 5 |- F/_ x B
5	4	nfcri 2342	. . . 4 |- F/ x z e. B
6	3, 5	nfan 1588	. . 3 |- F/ x ( y e. A /\ z e. B )
7	6	nfopab 4113	. 2 |- F/_ x { <. y , z >. | ( y e. A /\ z e. B ) }
8	1, 7	nfcxfr 2345	1 |- F/_ x ( A X. B )

Syntax hints:   /\ wa 104   e. wcel 2176   F/_wnfc 2335   {copab 4105   X. cxp 4674

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-opab 4107  df-xp 4682

This theorem is referenced by:  opeliunxp  4731  nfres  4962  mpomptsx  6285  dmmpossx  6287  fmpox  6288  disjxp1  6324  nfdju  7146  fsum2dlemstep  11778  fisumcom2  11782  fprod2dlemstep  11966  fprodcom2fi  11970