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Theorem nfopab2 5174
Description: The second abstraction variable in an ordered-pair class abstraction 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 5166 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 nfe1 2147 . . . 4 𝑦𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
32nfex 2318 . . 3 𝑦𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
43nfab 2911 . 2 𝑦{𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
51, 4nfcxfr 2903 1 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wex 1781  {cab 2714  wnfc 2885  cop 4590  {copab 5165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-opab 5166
This theorem is referenced by:  rexopabb  5483  ssopab2bw  5502  ssopab2b  5504  0nelopabOLD  5523  dmopab  5869  rnopab  5907  funopab  6533  fvopab5  6977  zfrep6  7879  opabdm  31359  opabrn  31360  fpwrelmap  31476  fineqvrep  33508  bj-opabco  35597  vvdifopab  36658  aomclem8  41297  areaquad  41459  sprsymrelf  45588
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