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Theorem nfsab1 2715
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove use of ax-12 2169. (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 2708 . 2 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
2 nfs1v 2151 . 2 𝑥[𝑦 / 𝑥]𝜑
31, 2nfxfr 1853 1 𝑥 𝑦 ∈ {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wnf 1783  [wsb 2065  wcel 2104  {cab 2707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-10 2135
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708
This theorem is referenced by:  hbab1  2716  abbib  2802  clelabOLD  2878  nfab1  2903  ralab2  3692  rexab2  3694  eluniab  4922  elintabOLD  4962  opabex3d  7954  opabex3rd  7955  opabex3  7956  setindtrs  42066  rababg  42627  scottabf  43301
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