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Theorem nfabd 2238
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
Hypotheses
Ref Expression
nfabd.1 𝑦𝜑
nfabd.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfabd (𝜑𝑥{𝑦𝜓})

Proof of Theorem nfabd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1462 . 2 𝑧𝜑
2 df-clab 2069 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
3 nfabd.1 . . . 4 𝑦𝜑
4 nfabd.2 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
53, 4nfsbd 1893 . . 3 (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
62, 5nfxfrd 1405 . 2 (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦𝜓})
71, 6nfcd 2215 1 (𝜑𝑥{𝑦𝜓})
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1390  wcel 1434  [wsb 1686  {cab 2068  wnfc 2207
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-nfc 2209
This theorem is referenced by:  nfsbcd  2835  nfcsb1d  2937  nfcsbd  2940  nfifd  3384  nfunid  3616  nfiotadxy  4900
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