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Theorem nfsab1 2722
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove use of ax-12 2177. (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 2715 . 2 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
2 nfs1v 2156 . 2 𝑥[𝑦 / 𝑥]𝜑
31, 2nfxfr 1853 1 𝑥 𝑦 ∈ {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wnf 1783  [wsb 2064  wcel 2108  {cab 2714
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 1910  ax-6 1967  ax-7 2007  ax-10 2141
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715
This theorem is referenced by:  hbab1  2723  abbib  2811  nfab1  2907  ralab2  3703  rexab2  3705  eluniab  4921  elintabOLD  4959  opabex3d  7990  opabex3rd  7991  opabex3  7992  setindtrs  43037  rababg  43587  scottabf  44259
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