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Theorem nfopab 3996
 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 3990 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 nfv 1508 . . . . . 6 𝑧 𝑤 = ⟨𝑥, 𝑦
3 nfopab.1 . . . . . 6 𝑧𝜑
42, 3nfan 1544 . . . . 5 𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
54nfex 1616 . . . 4 𝑧𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
65nfex 1616 . . 3 𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
76nfab 2286 . 2 𝑧{𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
81, 7nfcxfr 2278 1 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1331  Ⅎwnf 1436  ∃wex 1468  {cab 2125  Ⅎwnfc 2268  ⟨cop 3530  {copab 3988 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-opab 3990 This theorem is referenced by:  csbopabg  4006  nfmpt  4020  nfxp  4566  nfco  4704  nfcnv  4718  nfofr  5988
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