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Theorem rab2ex 5229
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 = {𝑦𝐴𝜓}
rab2ex.2 𝐴 ∈ V
Assertion
Ref Expression
rab2ex {𝑥𝐵𝜑} ∈ 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
StepHypRef Expression
1 rab2ex.1 . . 3 𝐵 = {𝑦𝐴𝜓}
2 rab2ex.2 . . 3 𝐴 ∈ V
31, 2rabex2 5228 . 2 𝐵 ∈ V
43rabex 5226 1 {𝑥𝐵𝜑} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-in 3940  df-ss 3949
This theorem is referenced by:  gsumbagdiag  20084  psrlidm  20111  psrridm  20112  psrass1  20113  mdegmullem  24599  vtxdginducedm1lem4  27251  vtxdginducedm1  27252
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