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Theorem nfxp 4371
 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 4351 . 2 (𝐴 × 𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
2 nfxp.1 . . . . 5 𝑥𝐴
32nfcri 2172 . . . 4 𝑥 𝑦𝐴
4 nfxp.2 . . . . 5 𝑥𝐵
54nfcri 2172 . . . 4 𝑥 𝑧𝐵
63, 5nfan 1457 . . 3 𝑥(𝑦𝐴𝑧𝐵)
76nfopab 3825 . 2 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
81, 7nfcxfr 2175 1 𝑥(𝐴 × 𝐵)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ∈ wcel 1393  Ⅎwnfc 2165  {copab 3817   × cxp 4343 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-opab 3819  df-xp 4351 This theorem is referenced by:  opeliunxp  4395  nfres  4614  mpt2mptsx  5823  dmmpt2ssx  5825  fmpt2x  5826
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