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Theorem nfsab1 2811
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove use of ax-12 2179. (Revised by SN, 20-Sep-2023.)
Assertion
Ref Expression
nfsab1 𝑥 𝑦 ∈ {𝑥𝜑}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem nfsab1
StepHypRef Expression
1 df-clab 2803 . 2 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
2 nfs1v 2161 . 2 𝑥[𝑦 / 𝑥]𝜑
31, 2nfxfr 1854 1 𝑥 𝑦 ∈ {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wnf 1785  [wsb 2070  wcel 2115  {cab 2802
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 1912  ax-6 1971  ax-7 2016  ax-10 2146
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803
This theorem is referenced by:  abbi  2891  clelab  2959  nfab1  2984  ralab2  3674  ralab2OLD  3675  rexab2  3677  rexab2OLD  3678  eluniab  4839  elintab  4873  opabex3d  7661  opabex3rd  7662  opabex3  7663  setindtrs  39882  rababg  40189  scottabf  40868
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