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Theorem nfopab2 4753
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 4746 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 nfe1 2067 . . . 4 𝑦𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
32nfex 2192 . . 3 𝑦𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
43nfab 2798 . 2 𝑦{𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
51, 4nfcxfr 2791 1 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wex 1744  {cab 2637  wnfc 2780  cop 4216  {copab 4745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-opab 4746
This theorem is referenced by:  opelopabsb  5014  ssopab2b  5031  dmopab  5367  rnopab  5402  funopab  5961  fvopab5  6349  0neqopab  6740  zfrep6  7176  opabdm  29549  opabrn  29550  fpwrelmap  29636  vvdifopab  34165  aomclem8  37948  areaquad  38119  sprsymrelf  42070
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