Theorem rabexf 45893

Description: Separation Scheme in terms of a restricted class abstraction. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
rabexf.1	⊢ Ⅎ𝑥𝐴
rabexf.2	⊢ 𝐴 ∈ 𝑉

Assertion

Ref	Expression
rabexf	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V

Proof of Theorem rabexf

Step	Hyp	Ref	Expression
1		rabexf.2	. 2 ⊢ 𝐴 ∈ 𝑉
2		rabexf.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	2	rabexgf 45785	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
4	1, 3	ax-mp 5	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V

This proof depends on syntax axioms:   ∈ wcel 2146  Ⅎwnfc 2913  {crab 3419  Vcvv 3458

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-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-rab 3420  df-v 3460  df-in 3915  df-ss 3925

This theorem is used by:  limsupequzmpt2  46473  liminfequzmpt2  46546  fsupdm  47597  finfdm  47601