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Theorem nfxp 5112
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 5090 . 2 (𝐴 × 𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
2 nfxp.1 . . . . 5 𝑥𝐴
32nfcri 2755 . . . 4 𝑥 𝑦𝐴
4 nfxp.2 . . . . 5 𝑥𝐵
54nfcri 2755 . . . 4 𝑥 𝑧𝐵
63, 5nfan 1825 . . 3 𝑥(𝑦𝐴𝑧𝐵)
76nfopab 4690 . 2 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
81, 7nfcxfr 2759 1 𝑥(𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 384  wcel 1987  wnfc 2748  {copab 4682   × cxp 5082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-opab 4684  df-xp 5090
This theorem is referenced by:  opeliunxp  5141  nfres  5368  mpt2mptsx  7193  dmmpt2ssx  7195  fmpt2x  7196  ovmptss  7218  axcc2  9219  fsum2dlem  14448  fsumcom2  14452  fsumcom2OLD  14453  fprod2dlem  14654  fprodcom2  14658  fprodcom2OLD  14659  gsumcom2  18314  prdsdsf  22112  prdsxmet  22114  aciunf1lem  29345  esum2dlem  29977  poimirlem16  33096  poimirlem19  33099  dvnprodlem1  39498  stoweidlem21  39575  stoweidlem47  39601  opeliun2xp  41429  dmmpt2ssx2  41433
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