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Theorem nfopab 5105
 Description: Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) Remove disjoint variable conditions. (Revised 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 5100 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 nfv 1916 . . . . . 6 𝑧 𝑤 = ⟨𝑥, 𝑦
3 nfopab.1 . . . . . 6 𝑧𝜑
42, 3nfan 1901 . . . . 5 𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
54nfex 2333 . . . 4 𝑧𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
65nfex 2333 . . 3 𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
76nfab 2926 . 2 𝑧{𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
81, 7nfcxfr 2918 1 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1539  ∃wex 1782  Ⅎwnf 1786  {cab 2736  Ⅎwnfc 2900  ⟨cop 4532  {copab 5099 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-opab 5100 This theorem is referenced by:  nfmpt  5134  csbopab  5417  csbopabgALT  5418  nfxp  5562  nfco  5712  nfcnv  5725  nfofr  7418
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