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Theorem rabex2 4171
Description: Separation Scheme in terms of a restricted class abstraction. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)
Hypotheses
Ref Expression
rabex2.1 𝐵 = {𝑥𝐴𝜓}
rabex2.2 𝐴 ∈ V
Assertion
Ref Expression
rabex2 𝐵 ∈ V
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem rabex2
StepHypRef Expression
1 rabex2.2 . 2 𝐴 ∈ V
2 rabex2.1 . . 3 𝐵 = {𝑥𝐴𝜓}
3 id 19 . . 3 (𝐴 ∈ V → 𝐴 ∈ V)
42, 3rabexd 4170 . 2 (𝐴 ∈ V → 𝐵 ∈ V)
51, 4ax-mp 5 1 𝐵 ∈ V
Colors of variables: wff set class
Syntax hints:   = wceq 1364  wcel 2160  {crab 2472  Vcvv 2756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-sep 4143
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2758  df-in 3155  df-ss 3162
This theorem is referenced by:  rab2ex  4172
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