Theorem rab2ex 5328

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 5327	. 2 ⊢ 𝐵 ∈ V
4	3	rabex 5325	1 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} ∈ V

Syntax hints:   = wceq 1533   ∈ wcel 2098  {crab 3426  Vcvv 3468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-in 3950  df-ss 3960

This theorem is referenced by:  gsumbagdiagOLD  21829  gsumbagdiag  21832  psrlidm  21861  psrridm  21862  psrass1  21863  mdegmullem  25965  vtxdginducedm1lem4  29304  vtxdginducedm1  29305