Theorem rab2ex 5254

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 5253	. 2 ⊢ 𝐵 ∈ V
4	3	rabex 5251	1 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} ∈ V

Syntax hints:   = wceq 1539   ∈ wcel 2108  {crab 3067  Vcvv 3422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-in 3890  df-ss 3900

This theorem is referenced by:  gsumbagdiagOLD  21052  gsumbagdiag  21055  psrlidm  21082  psrridm  21083  psrass1  21084  mdegmullem  25148  vtxdginducedm1lem4  27812  vtxdginducedm1  27813