Theorem nfsab1 2752

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove use of ax-12 2216. (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 2745	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
2		nfs1v 2194	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3	1, 2	nfxfr 1886	1 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}

This proof depends on syntax axioms:  Ⅎwnf 1816  [wsb 2099   ∈ wcel 2146  {cab 2744

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745

This theorem is used by:  hbab1  2753  abbib  2835  nfab1  2930  ralab2  3663  rexab2  3665  eluniab  4891  opabex3d  7971  opabex3rd  7972  opabex3  7973  scottabf  9878  setindtrs  43793  rababg  44341