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Theorem rab2ex 5348
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 5347 . 2 𝐵 ∈ V
43rabex 5345 1 {𝑥𝐵𝜑} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-pw 4607
This theorem is referenced by:  gsumbagdiag  21969  psrlidm  22000  psrridm  22001  psrass1  22002  mdegmullem  26132  vtxdginducedm1lem4  29575  vtxdginducedm1  29576
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