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Theorem nfopab1 4681
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 4674 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 nfe1 2024 . . 3 𝑥𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
32nfab 2765 . 2 𝑥{𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
41, 3nfcxfr 2759 1 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wex 1701  {cab 2607  wnfc 2748  cop 4154  {copab 4672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-opab 4674
This theorem is referenced by:  nfmpt1  4707  opelopabsb  4945  ssopab2b  4962  dmopab  5295  rnopab  5330  funopab  5881  fvopab5  6265  0neqopab  6651  zfrep6  7081  opabdm  29263  opabrn  29264  fpwrelmap  29348  aomclem8  37108  sprsymrelf  41030
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