Theorem nfsab1 2722

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove use of ax-12 2178. (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 2715	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
2		nfs1v 2157	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3	1, 2	nfxfr 1853	1 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}

Syntax hints:  Ⅎwnf 1783  [wsb 2065   ∈ wcel 2109  {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 2008  ax-10 2142

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715

This theorem is referenced by:  hbab1  2723  abbib  2805  nfab1  2901  ralab2  3685  rexab2  3687  eluniab  4902  elintabOLD  4940  opabex3d  7969  opabex3rd  7970  opabex3  7971  setindtrs  43016  rababg  43565  scottabf  44231