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Theorem nfoprab3 5556
 Description: The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.)
Assertion
Ref Expression
nfoprab3 𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Proof of Theorem nfoprab3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 df-oprab 5516 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
2 nfe1 1385 . . . . 5 𝑧𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
32nfex 1528 . . . 4 𝑧𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
43nfex 1528 . . 3 𝑧𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
54nfab 2182 . 2 𝑧{𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
61, 5nfcxfr 2175 1 𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1243  ∃wex 1381  {cab 2026  Ⅎwnfc 2165  ⟨cop 3378  {coprab 5513 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-oprab 5516 This theorem is referenced by:  ssoprab2b  5562  ovi3  5637  tposoprab  5895
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