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Theorem nfsab 2032
 Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
nfsab.1 𝑥𝜑
Assertion
Ref Expression
nfsab 𝑥 𝑧 ∈ {𝑦𝜑}
Distinct variable group:   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem nfsab
StepHypRef Expression
1 nfsab.1 . . . 4 𝑥𝜑
21nfri 1412 . . 3 (𝜑 → ∀𝑥𝜑)
32hbab 2031 . 2 (𝑧 ∈ {𝑦𝜑} → ∀𝑥 𝑧 ∈ {𝑦𝜑})
43nfi 1351 1 𝑥 𝑧 ∈ {𝑦𝜑}
 Colors of variables: wff set class Syntax hints:  Ⅎwnf 1349   ∈ wcel 1393  {cab 2026 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 This theorem is referenced by:  nfab  2182  peano2  4318
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