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Theorem bdcrab 12729
Description: A class defined by restricted abstraction from a bounded class and a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
Hypotheses
Ref Expression
bdcrab.1 BOUNDED 𝐴
bdcrab.2 BOUNDED 𝜑
Assertion
Ref Expression
bdcrab BOUNDED {𝑥𝐴𝜑}
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bdcrab
StepHypRef Expression
1 bdcrab.1 . . . . 5 BOUNDED 𝐴
21bdeli 12723 . . . 4 BOUNDED 𝑥𝐴
3 bdcrab.2 . . . 4 BOUNDED 𝜑
42, 3ax-bdan 12692 . . 3 BOUNDED (𝑥𝐴𝜑)
54bdcab 12726 . 2 BOUNDED {𝑥 ∣ (𝑥𝐴𝜑)}
6 df-rab 2397 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
75, 6bdceqir 12721 1 BOUNDED {𝑥𝐴𝜑}
Colors of variables: wff set class
Syntax hints:  wa 103  wcel 1461  {cab 2099  {crab 2392  BOUNDED wbd 12689  BOUNDED wbdc 12717
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1404  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-4 1468  ax-17 1487  ax-ial 1495  ax-ext 2095  ax-bd0 12690  ax-bdan 12692  ax-bdsb 12699
This theorem depends on definitions:  df-bi 116  df-clab 2100  df-cleq 2106  df-clel 2109  df-rab 2397  df-bdc 12718
This theorem is referenced by:  bdrabexg  12783
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