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Theorem nfmpt21 5697
Description: Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
Assertion
Ref Expression
nfmpt21 𝑥(𝑥𝐴, 𝑦𝐵𝐶)

Proof of Theorem nfmpt21
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-mpt2 5639 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
2 nfoprab1 5680 . 2 𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
31, 2nfcxfr 2225 1 𝑥(𝑥𝐴, 𝑦𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1289  wcel 1438  wnfc 2215  {coprab 5635  cmpt2 5636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-oprab 5638  df-mpt2 5639
This theorem is referenced by:  ovmpt2s  5750  ov2gf  5751  ovmpt2dxf  5752  ovmpt2df  5758  ovmpt2dv2  5760  xpcomco  6522  mapxpen  6544
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