Theorem rab2ex 5287

Description: A class abstraction based on a class abstraction based on a set is a set. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)

Hypotheses

Ref	Expression
rab2ex.1	⊢ 𝐵 = {𝑦 ∈ 𝐴 ∣ 𝜓}
rab2ex.2	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
rab2ex	⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} ∈ V

Distinct variable groups:   𝑥,𝐵   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem rab2ex

Step	Hyp	Ref	Expression
1		rab2ex.1	. . 3 ⊢ 𝐵 = {𝑦 ∈ 𝐴 ∣ 𝜓}
2		rab2ex.2	. . 3 ⊢ 𝐴 ∈ V
3	1, 2	rabex2 5286	. 2 ⊢ 𝐵 ∈ V
4	3	rabex 5284	1 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} ∈ V

Syntax hints:   = wceq 1541   ∈ wcel 2113  {crab 3399  Vcvv 3440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-in 3908  df-ss 3918  df-pw 4556

This theorem is referenced by:  gsumbagdiag  21887  psrlidm  21917  psrridm  21918  psrass1  21919  mdegmullem  26039  vtxdginducedm1lem4  29616  vtxdginducedm1  29617  mplmulmvr  33704