Theorem nfxp 4389

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 4369	. 2 |- ( A X. B ) = { <. y , z >. | ( y e. A /\ z e. B ) }
2		nfxp.1	. . . . 5 |- F/_ x A
3	2	nfcri 2213	. . . 4 |- F/ x y e. A
4		nfxp.2	. . . . 5 |- F/_ x B
5	4	nfcri 2213	. . . 4 |- F/ x z e. B
6	3, 5	nfan 1497	. . 3 |- F/ x ( y e. A /\ z e. B )
7	6	nfopab 3846	. 2 |- F/_ x { <. y , z >. | ( y e. A /\ z e. B ) }
8	1, 7	nfcxfr 2216	1 |- F/_ x ( A X. B )

Syntax hints:   /\ wa 102   e. wcel 1433   F/_wnfc 2206   {copab 3838   X. cxp 4361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-opab 3840  df-xp 4369

This theorem is referenced by:  opeliunxp  4413  nfres  4632  mpt2mptsx  5843  dmmpt2ssx  5845  fmpt2x  5846