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Theorem nfxp 4566
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 4545 . 2 (𝐴 × 𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
2 nfxp.1 . . . . 5 𝑥𝐴
32nfcri 2275 . . . 4 𝑥 𝑦𝐴
4 nfxp.2 . . . . 5 𝑥𝐵
54nfcri 2275 . . . 4 𝑥 𝑧𝐵
63, 5nfan 1544 . . 3 𝑥(𝑦𝐴𝑧𝐵)
76nfopab 3996 . 2 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
81, 7nfcxfr 2278 1 𝑥(𝐴 × 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 103  wcel 1480  wnfc 2268  {copab 3988   × cxp 4537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-opab 3990  df-xp 4545
This theorem is referenced by:  opeliunxp  4594  nfres  4821  mpomptsx  6095  dmmpossx  6097  fmpox  6098  disjxp1  6133  nfdju  6927  fsum2dlemstep  11215  fisumcom2  11219
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