Theorem nfxp 4810

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/_xA
nfxp.2	|- F/_xB

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 4784	. 2 |- (A X. B) = {<.y, z>. | (y e. A /\ z e. B)}
2		nfxp.1	. . . . 5 |- F/_xA
3	2	nfcri 2483	. . . 4 |- F/x y e. A
4		nfxp.2	. . . . 5 |- F/_xB
5	4	nfcri 2483	. . . 4 |- F/x z e. B
6	3, 5	nfan 1824	. . 3 |- F/x(y e. A /\ z e. B)
7	6	nfopab 4627	. 2 |- F/_x{<.y, z>. | (y e. A /\ z e. B)}
8	1, 7	nfcxfr 2486	1 |- F/_x(A X. B)

Syntax hints:   /\ wa 358   e. wcel 1710   F/_wnfc 2476  {copab 4622   X. cxp 4770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-opab 4623  df-xp 4784

This theorem is referenced by:  opeliunxp  4820  nfres  4936  fmpt2x  5730