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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 xA
nfxp.2 xB
Assertion
Ref Expression
nfxp x(A × B)

Proof of Theorem nfxp
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xp 4784 . 2 (A × B) = {y, z (y A z B)}
2 nfxp.1 . . . . 5 xA
32nfcri 2483 . . . 4 x y A
4 nfxp.2 . . . . 5 xB
54nfcri 2483 . . . 4 x z B
63, 5nfan 1824 . . 3 x(y A z B)
76nfopab 4627 . 2 x{y, z (y A z B)}
81, 7nfcxfr 2486 1 x(A × B)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   ∈ wcel 1710  Ⅎwnfc 2476  {copab 4622   × cxp 4770 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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
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