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Theorem nfoprab 5926
Description: Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.)
Hypothesis
Ref Expression
nfoprab.1 𝑤𝜑
Assertion
Ref Expression
nfoprab 𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
Distinct variable groups:   𝑥,𝑤   𝑦,𝑤   𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem nfoprab
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 df-oprab 5878 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
2 nfv 1528 . . . . . . 7 𝑤 𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧
3 nfoprab.1 . . . . . . 7 𝑤𝜑
42, 3nfan 1565 . . . . . 6 𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
54nfex 1637 . . . . 5 𝑤𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
65nfex 1637 . . . 4 𝑤𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
76nfex 1637 . . 3 𝑤𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
87nfab 2324 . 2 𝑤{𝑣 ∣ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
91, 8nfcxfr 2316 1 𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wnf 1460  wex 1492  {cab 2163  wnfc 2306  cop 3595  {coprab 5875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-oprab 5878
This theorem is referenced by:  nfmpo  5943
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