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Theorem nfopab 3815
 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 3809 . 2 {⟨x, y⟩ ∣ φ} = {wxy(w = ⟨x, y φ)}
2 nfv 1418 . . . . . 6 z w = ⟨x, y
3 nfopab.1 . . . . . 6 zφ
42, 3nfan 1454 . . . . 5 z(w = ⟨x, y φ)
54nfex 1525 . . . 4 zy(w = ⟨x, y φ)
65nfex 1525 . . 3 zxy(w = ⟨x, y φ)
76nfab 2179 . 2 z{wxy(w = ⟨x, y φ)}
81, 7nfcxfr 2172 1 z{⟨x, y⟩ ∣ φ}
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1242  Ⅎwnf 1346  ∃wex 1378  {cab 2023  Ⅎwnfc 2162  ⟨cop 3369  {copab 3807 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-opab 3809 This theorem is referenced by:  csbopabg  3825  nfmpt  3839  nfxp  4313  nfco  4443  nfcnv  4456
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