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Theorem nfsab 2169
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 1519 . . 3 (𝜑 → ∀𝑥𝜑)
32hbab 2168 . 2 (𝑧 ∈ {𝑦𝜑} → ∀𝑥 𝑧 ∈ {𝑦𝜑})
43nfi 1462 1 𝑥 𝑧 ∈ {𝑦𝜑}
Colors of variables: wff set class
Syntax hints:  wnf 1460  wcel 2148  {cab 2163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164
This theorem is referenced by:  nfab  2324  peano2  4594
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