Theorem nfsab1 2746

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove use of ax-12 2212. (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 2739	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
2		nfs1v 2192	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3	1, 2	nfxfr 1885	1 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}

This proof depends on syntax axioms:  Ⅎwnf 1815  [wsb 2098   ∈ wcel 2144  {cab 2738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739

This theorem is used by:  hbab1  2747  abbib  2829  nfab1  2924  ralab2  3654  rexab2  3656  eluniab  4880  opabex3d  7977  opabex3rd  7978  opabex3  7979  scottabf  9942  setindtrs  44125  rababg  44668