Theorem nfsab1 2723

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove use of ax-12 2173. (Revised by SN, 20-Sep-2023.)

Assertion

Ref	Expression
nfsab1	⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem nfsab1

Step	Hyp	Ref	Expression
1		df-clab 2716	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
2		nfs1v 2155	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3	1, 2	nfxfr 1856	1 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}

Syntax hints:  Ⅎwnf 1787  [wsb 2068   ∈ wcel 2108  {cab 2715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716

This theorem is referenced by:  hbab1  2724  abbi  2811  clelabOLD  2883  nfab1  2908  ralab2  3627  ralab2OLD  3628  rexab2  3630  rexab2OLD  3631  eluniab  4851  elintab  4887  opabex3d  7781  opabex3rd  7782  opabex3  7783  setindtrs  40763  rababg  41070  scottabf  41747