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Theorem nfopab 4912
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 4907 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 nfv 2005 . . . . . 6 𝑧 𝑤 = ⟨𝑥, 𝑦
3 nfopab.1 . . . . . 6 𝑧𝜑
42, 3nfan 1990 . . . . 5 𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
54nfex 2330 . . . 4 𝑧𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
65nfex 2330 . . 3 𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
76nfab 2953 . 2 𝑧{𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
81, 7nfcxfr 2946 1 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1637  wex 1859  wnf 1863  {cab 2792  wnfc 2935  cop 4376  {copab 4906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-opab 4907
This theorem is referenced by:  nfmpt  4940  csbopab  5203  csbopabgALT  5204  nfxp  5343  nfco  5489  nfcnv  5502  nfofr  7133
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