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Theorem rab2ex 5335
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 5334 . 2 𝐵 ∈ V
43rabex 5332 1 {𝑥𝐵𝜑} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2105  {crab 3431  Vcvv 3473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-in 3955  df-ss 3965
This theorem is referenced by:  gsumbagdiagOLD  21802  gsumbagdiag  21805  psrlidm  21834  psrridm  21835  psrass1  21836  mdegmullem  25934  vtxdginducedm1lem4  29232  vtxdginducedm1  29233
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