Theorem rab2ex 5337

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 5336	. 2 ⊢ 𝐵 ∈ V
4	3	rabex 5334	1 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} ∈ V

Syntax hints:   = wceq 1534   ∈ wcel 2099  {crab 3429  Vcvv 3471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-in 3954  df-ss 3964

This theorem is referenced by:  gsumbagdiagOLD  21873  gsumbagdiag  21876  psrlidm  21905  psrridm  21906  psrass1  21907  mdegmullem  26027  vtxdginducedm1lem4  29369  vtxdginducedm1  29370