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Theorem nfsab 2748
 Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Add disjoint variable condition to avoid ax-13 2379. See nfsabg 2749 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 2193 . . 3 (𝜑 → ∀𝑥𝜑)
32hbab 2746 . 2 (𝑧 ∈ {𝑦𝜑} → ∀𝑥 𝑧 ∈ {𝑦𝜑})
43nf5i 2147 1 𝑥 𝑧 ∈ {𝑦𝜑}
 Colors of variables: wff setvar class Syntax hints:  Ⅎwnf 1785   ∈ wcel 2111  {cab 2735 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736 This theorem is referenced by:  nfab  2925  upbdrech  42305  ssfiunibd  42309
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