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Theorem nfsab 2809
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Add disjoint variable condition to avoid ax-13 2381. See nfsabg 2810 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Hypothesis
Ref Expression
nfsab.1 𝑥𝜑
Assertion
Ref Expression
nfsab 𝑥 𝑧 ∈ {𝑦𝜑}
Distinct variable groups:   𝑥,𝑦   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem nfsab
StepHypRef Expression
1 nfsab.1 . . . 4 𝑥𝜑
21nf5ri 2185 . . 3 (𝜑 → ∀𝑥𝜑)
32hbab 2807 . 2 (𝑧 ∈ {𝑦𝜑} → ∀𝑥 𝑧 ∈ {𝑦𝜑})
43nf5i 2141 1 𝑥 𝑧 ∈ {𝑦𝜑}
Colors of variables: wff setvar class
Syntax hints:  wnf 1775  wcel 2105  {cab 2796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-11 2151  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797
This theorem is referenced by:  nfab  2981  upbdrech  41448  ssfiunibd  41452
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