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Theorem nfopab 3853
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 3847 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 nfv 1437 . . . . . 6 𝑧 𝑤 = ⟨𝑥, 𝑦
3 nfopab.1 . . . . . 6 𝑧𝜑
42, 3nfan 1473 . . . . 5 𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
54nfex 1544 . . . 4 𝑧𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
65nfex 1544 . . 3 𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
76nfab 2198 . 2 𝑧{𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
81, 7nfcxfr 2191 1 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  wnf 1365  wex 1397  {cab 2042  wnfc 2181  cop 3406  {copab 3845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-opab 3847
This theorem is referenced by:  csbopabg  3863  nfmpt  3877  nfxp  4399  nfco  4529  nfcnv  4542
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