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Theorem nfxp 5717
Description: Bound-variable hypothesis builder for Cartesian 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 5690 . 2 (𝐴 × 𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
2 nfxp.1 . . . . 5 𝑥𝐴
32nfcri 2896 . . . 4 𝑥 𝑦𝐴
4 nfxp.2 . . . . 5 𝑥𝐵
54nfcri 2896 . . . 4 𝑥 𝑧𝐵
63, 5nfan 1898 . . 3 𝑥(𝑦𝐴𝑧𝐵)
76nfopab 5211 . 2 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
81, 7nfcxfr 2902 1 𝑥(𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 395  wcel 2107  wnfc 2889  {copab 5204   × cxp 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-opab 5205  df-xp 5690
This theorem is referenced by:  opeliunxp  5751  opeliun2xp  5752  nfres  5998  mpomptsx  8090  dmmpossx  8092  fmpox  8093  ovmptss  8119  nfdju  9948  axcc2  10478  fsum2dlem  15807  fsumcom2  15811  fprod2dlem  16017  fprodcom2  16021  gsumcom2  19994  prdsdsf  24378  prdsxmet  24380  iunxpssiun1  32582  djussxp2  32659  aciunf1lem  32673  gsumpart  33061  esum2dlem  34094  poimirlem16  37644  poimirlem19  37647  dvnprodlem1  45966  stoweidlem21  46041  stoweidlem47  46067  dmmpossx2  48258
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