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Theorem nfopab 4685
 Description: Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 11-Jul-2011.)
Hypothesis
Ref Expression
nfopab.1 𝑧𝜑
Assertion
Ref Expression
nfopab 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem nfopab
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 df-opab 4679 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 nfv 1840 . . . . . 6 𝑧 𝑤 = ⟨𝑥, 𝑦
3 nfopab.1 . . . . . 6 𝑧𝜑
42, 3nfan 1825 . . . . 5 𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
54nfex 2151 . . . 4 𝑧𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
65nfex 2151 . . 3 𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
76nfab 2765 . 2 𝑧{𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
81, 7nfcxfr 2759 1 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 384   = wceq 1480  ∃wex 1701  Ⅎwnf 1705  {cab 2607  Ⅎwnfc 2748  ⟨cop 4159  {copab 4677 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 4679 This theorem is referenced by:  nfmpt  4711  csbopab  4973  csbopabgALT  4974  nfxp  5107  nfco  5252  nfcnv  5266
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