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Theorem nfopab 4627
 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 4623 . 2 {x, y φ} = {w xy(w = x, y φ)}
2 nfv 1619 . . . . . 6 z w = x, y
3 nfopab.1 . . . . . 6 zφ
42, 3nfan 1824 . . . . 5 z(w = x, y φ)
54nfex 1843 . . . 4 zy(w = x, y φ)
65nfex 1843 . . 3 zxy(w = x, y φ)
76nfab 2493 . 2 z{w xy(w = x, y φ)}
81, 7nfcxfr 2486 1 z{x, y φ}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  {cab 2339  Ⅎwnfc 2476  ⟨cop 4561  {copab 4622 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-opab 4623 This theorem is referenced by:  csbopabg  4637  nfxp  4810  nfco  4882  nfcnv  4891  nfmpt  5671
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