Theorem nfsab1 2720

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove use of ax-12 2175. (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 2713	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
2		nfs1v 2154	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3	1, 2	nfxfr 1850	1 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}

Syntax hints:  Ⅎwnf 1780  [wsb 2062   ∈ wcel 2106  {cab 2712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713

This theorem is referenced by:  hbab1  2721  abbib  2809  nfab1  2905  ralab2  3706  rexab2  3708  eluniab  4926  elintabOLD  4964  opabex3d  7989  opabex3rd  7990  opabex3  7991  setindtrs  43014  rababg  43564  scottabf  44236