Theorem nfxp 4796

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 4775	. 2 |- ( A X. B ) = { <. y , z >. | ( y e. A /\ z e. B ) }
2		nfxp.1	. . . . 5 |- F/_ x A
3	2	nfcri 2386	. . . 4 |- F/ x y e. A
4		nfxp.2	. . . . 5 |- F/_ x B
5	4	nfcri 2386	. . . 4 |- F/ x z e. B
6	3, 5	nfan 1618	. . 3 |- F/ x ( y e. A /\ z e. B )
7	6	nfopab 4194	. 2 |- F/_ x { <. y , z >. | ( y e. A /\ z e. B ) }
8	1, 7	nfcxfr 2389	1 |- F/_ x ( A X. B )

Syntax hints:   /\ wa 104   e. wcel 2209   F/_wnfc 2379   {copab 4186   X. cxp 4767

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-opab 4188  df-xp 4775

This theorem is referenced by:  opeliunxp  4825  nfres  5060  mpomptsx  6423  dmmpossx  6425  fmpox  6426  disjxp1  6462  nfdju  7372  fsum2dlemstep  12179  fisumcom2  12183  fprod2dlemstep  12367  fprodcom2fi  12371