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Theorem nfxp 5176
 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 5149 . 2 (𝐴 × 𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
2 nfxp.1 . . . . 5 𝑥𝐴
32nfcri 2787 . . . 4 𝑥 𝑦𝐴
4 nfxp.2 . . . . 5 𝑥𝐵
54nfcri 2787 . . . 4 𝑥 𝑧𝐵
63, 5nfan 1868 . . 3 𝑥(𝑦𝐴𝑧𝐵)
76nfopab 4751 . 2 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
81, 7nfcxfr 2791 1 𝑥(𝐴 × 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   ∈ wcel 2030  Ⅎwnfc 2780  {copab 4745   × cxp 5141 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-opab 4746  df-xp 5149 This theorem is referenced by:  opeliunxp  5204  nfres  5430  mpt2mptsx  7278  dmmpt2ssx  7280  fmpt2x  7281  ovmptss  7303  axcc2  9297  fsum2dlem  14545  fsumcom2  14549  fsumcom2OLD  14550  fprod2dlem  14754  fprodcom2  14758  fprodcom2OLD  14759  gsumcom2  18420  prdsdsf  22219  prdsxmet  22221  aciunf1lem  29590  esum2dlem  30282  poimirlem16  33555  poimirlem19  33558  dvnprodlem1  40479  stoweidlem21  40556  stoweidlem47  40582  opeliun2xp  42436  dmmpt2ssx2  42440
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