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Theorem nfxp 5066
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 5044 . 2 (𝐴 × 𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
2 nfxp.1 . . . . 5 𝑥𝐴
32nfcri 2745 . . . 4 𝑥 𝑦𝐴
4 nfxp.2 . . . . 5 𝑥𝐵
54nfcri 2745 . . . 4 𝑥 𝑧𝐵
63, 5nfan 1816 . . 3 𝑥(𝑦𝐴𝑧𝐵)
76nfopab 4650 . 2 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
81, 7nfcxfr 2749 1 𝑥(𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 383  wcel 1977  wnfc 2738  {copab 4642   × cxp 5036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-opab 4644  df-xp 5044
This theorem is referenced by:  opeliunxp  5093  nfres  5319  mpt2mptsx  7122  dmmpt2ssx  7124  fmpt2x  7125  ovmptss  7145  axcc2  9142  fsum2dlem  14343  fsumcom2  14347  fsumcom2OLD  14348  fprod2dlem  14549  fprodcom2  14553  fprodcom2OLD  14554  gsumcom2  18197  prdsdsf  21982  prdsxmet  21984  aciunf1lem  28844  esum2dlem  29481  poimirlem16  32595  poimirlem19  32598  dvnprodlem1  38836  stoweidlem21  38914  stoweidlem47  38940  opeliun2xp  41904  dmmpt2ssx2  41908
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