Theorem zfausab 5260

Description: Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)

Hypothesis

Ref	Expression
zfausab.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
zfausab	⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem zfausab

Step	Hyp	Ref	Expression
1		zfausab.1	. 2 ⊢ 𝐴 ∈ V
2		ssab2 4010	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
3	1, 2	ssexi 5250	1 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V

Syntax hints:   ∧ wa 396   ∈ wcel 2119  {cab 2717  Vcvv 3431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218

This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1094  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-in 3890  df-ss 3900

This theorem is referenced by:  rabfmpunirn  32745