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Theorem nfoprab 5496
 Description: Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.)
Hypothesis
Ref Expression
nfoprab.1 wφ
Assertion
Ref Expression
nfoprab w{⟨⟨x, y⟩, z⟩ ∣ φ}
Distinct variable groups:   x,w   y,w   z,w
Allowed substitution hints:   φ(x,y,z,w)

Proof of Theorem nfoprab
Dummy variable v is distinct from all other variables.
StepHypRef Expression
1 df-oprab 5456 . 2 {⟨⟨x, y⟩, z⟩ ∣ φ} = {vxyz(v = ⟨⟨x, y⟩, z φ)}
2 nfv 1418 . . . . . . 7 w v = ⟨⟨x, y⟩, z
3 nfoprab.1 . . . . . . 7 wφ
42, 3nfan 1454 . . . . . 6 w(v = ⟨⟨x, y⟩, z φ)
54nfex 1525 . . . . 5 wz(v = ⟨⟨x, y⟩, z φ)
65nfex 1525 . . . 4 wyz(v = ⟨⟨x, y⟩, z φ)
76nfex 1525 . . 3 wxyz(v = ⟨⟨x, y⟩, z φ)
87nfab 2179 . 2 w{vxyz(v = ⟨⟨x, y⟩, z φ)}
91, 8nfcxfr 2172 1 w{⟨⟨x, y⟩, z⟩ ∣ φ}
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1242  Ⅎwnf 1346  ∃wex 1378  {cab 2023  Ⅎwnfc 2162  ⟨cop 3369  {coprab 5453 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-oprab 5456 This theorem is referenced by:  nfmpt2  5512
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