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Theorem nfopab2 4652
Description: The second abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
nfopab2 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Proof of Theorem nfopab2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-opab 4644 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 nfe1 2014 . . . 4 𝑦𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
32nfex 2140 . . 3 𝑦𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
43nfab 2755 . 2 𝑦{𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
51, 4nfcxfr 2749 1 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1475  wex 1695  {cab 2596  wnfc 2738  cop 4131  {copab 4642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-opab 4644
This theorem is referenced by:  opelopabsb  4910  ssopab2b  4927  dmopab  5257  rnopab  5291  funopab  5837  fvopab5  6217  0neqopab  6596  zfrep6  7027  opabdm  28803  opabrn  28804  fpwrelmap  28896  aomclem8  36649  areaquad  36821
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