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Theorem bdcrab 9241
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 {x Aφ}
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem bdcrab
StepHypRef Expression
1 bdcrab.1 . . . . 5 BOUNDED A
21bdeli 9235 . . . 4 BOUNDED x A
3 bdcrab.2 . . . 4 BOUNDED φ
42, 3ax-bdan 9204 . . 3 BOUNDED (x A φ)
54bdcab 9238 . 2 BOUNDED {x ∣ (x A φ)}
6 df-rab 2309 . 2 {x Aφ} = {x ∣ (x A φ)}
75, 6bdceqir 9233 1 BOUNDED {x Aφ}
Colors of variables: wff set class
Syntax hints:   wa 97   wcel 1390  {cab 2023  {crab 2304  BOUNDED wbd 9201  BOUNDED wbdc 9229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019  ax-bd0 9202  ax-bdan 9204  ax-bdsb 9211
This theorem depends on definitions:  df-bi 110  df-clab 2024  df-cleq 2030  df-clel 2033  df-rab 2309  df-bdc 9230
This theorem is referenced by:  bdrabexg  9291
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