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Theorem nfabd 2196
 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 1421 . 2 𝑧𝜑
2 df-clab 2027 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
3 nfabd.1 . . . 4 𝑦𝜑
4 nfabd.2 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
53, 4nfsbd 1851 . . 3 (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
62, 5nfxfrd 1364 . 2 (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦𝜓})
71, 6nfcd 2173 1 (𝜑𝑥{𝑦𝜓})
 Colors of variables: wff set class Syntax hints:   → wi 4  Ⅎwnf 1349   ∈ wcel 1393  [wsb 1645  {cab 2026  Ⅎwnfc 2165 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-nfc 2167 This theorem is referenced by:  nfsbcd  2783  nfcsb1d  2880  nfcsbd  2883  nfifd  3355  nfunid  3587  nfiotadxy  4870
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