Theorem rab2ex 5310

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 5309	. 2 ⊢ 𝐵 ∈ V
4	3	rabex 5307	1 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} ∈ V

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-pw 4562

This theorem is used by:  gsumbagdiag  22295  psrlidm  22324  psrridm  22325  psrass1  22326  mdegmullem  26451  vtxdginducedm1lem4  30180  vtxdginducedm1  30181  mplmulmvr  34282