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Theorem nfsab1 2718
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove use of ax-12 2172. (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 2711 . 2 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
2 nfs1v 2154 . 2 𝑥[𝑦 / 𝑥]𝜑
31, 2nfxfr 1856 1 𝑥 𝑦 ∈ {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wnf 1786  [wsb 2068  wcel 2107  {cab 2710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711
This theorem is referenced by:  hbab1  2719  abbib  2805  clelabOLD  2881  nfab1  2906  ralab2  3659  rexab2  3661  eluniab  4884  elintabOLD  4924  opabex3d  7902  opabex3rd  7903  opabex3  7904  setindtrs  41396  rababg  41938  scottabf  42612
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