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Theorem nfopab1 4854
Description: The first 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
nfopab1 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Proof of Theorem nfopab1
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-opab 4848 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 nfe1 2183 . . 3 𝑥𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
32nfab 2918 . 2 𝑥{𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
41, 3nfcxfr 2911 1 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  wex 1852  {cab 2757  wnfc 2900  cop 4323  {copab 4847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-opab 4848
This theorem is referenced by:  nfmpt1  4882  opelopabsb  5119  ssopab2b  5136  dmopab  5472  rnopab  5507  funopab  6065  fvopab5  6454  0neqopab  6849  zfrep6  7285  opabdm  29761  opabrn  29762  fpwrelmap  29848  vvdifopab  34365  aomclem8  38155  sprsymrelf  42268
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