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Theorem nfopab2 4879
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 4872 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 nfe1 2192 . . . 4 𝑦𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
32nfex 2327 . . 3 𝑦𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
43nfab 2912 . 2 𝑦{𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
51, 4nfcxfr 2905 1 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1652  wex 1874  {cab 2751  wnfc 2894  cop 4340  {copab 4871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-opab 4872
This theorem is referenced by:  opelopabsb  5146  ssopab2b  5163  dmopab  5503  rnopab  5539  funopab  6103  fvopab5  6499  0neqopab  6896  zfrep6  7332  opabdm  29871  opabrn  29872  fpwrelmap  29957  vvdifopab  34459  aomclem8  38308  areaquad  38478  sprsymrelf  42414
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