MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfoprab3 Structured version   Visualization version   GIF version

Theorem nfoprab3 6691
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 6639 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
2 nfe1 2025 . . . . 5 𝑧𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
32nfex 2152 . . . 4 𝑧𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
43nfex 2152 . . 3 𝑧𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
54nfab 2766 . 2 𝑧{𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
61, 5nfcxfr 2760 1 𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1481  wex 1702  {cab 2606  wnfc 2749  cop 4174  {coprab 6636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-oprab 6639
This theorem is referenced by:  ssoprab2b  6697  ov3  6782  tposoprab  7373
  Copyright terms: Public domain W3C validator