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Theorem nfsab1 2717
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove use of ax-12 2171. (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 2710 . 2 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
2 nfs1v 2153 . 2 𝑥[𝑦 / 𝑥]𝜑
31, 2nfxfr 1855 1 𝑥 𝑦 ∈ {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wnf 1785  [wsb 2067  wcel 2106  {cab 2709
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 1913  ax-6 1971  ax-7 2011  ax-10 2137
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710
This theorem is referenced by:  hbab1  2718  abbib  2804  clelabOLD  2880  nfab1  2905  ralab2  3693  rexab2  3695  eluniab  4923  elintabOLD  4963  opabex3d  7951  opabex3rd  7952  opabex3  7953  setindtrs  41754  rababg  42315  scottabf  42989
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