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Theorem bdrabexg 12938
 Description: Bounded version of rabexg 4039. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bdrabexg.bd BOUNDED 𝜑
bdrabexg.bdc BOUNDED 𝐴
Assertion
Ref Expression
bdrabexg (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem bdrabexg
StepHypRef Expression
1 ssrab2 3150 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
2 bdrabexg.bdc . . . 4 BOUNDED 𝐴
3 bdrabexg.bd . . . 4 BOUNDED 𝜑
42, 3bdcrab 12884 . . 3 BOUNDED {𝑥𝐴𝜑}
54bdssexg 12936 . 2 (({𝑥𝐴𝜑} ⊆ 𝐴𝐴𝑉) → {𝑥𝐴𝜑} ∈ V)
61, 5mpan 418 1 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1463  {crab 2395  Vcvv 2658   ⊆ wss 3039  BOUNDED wbd 12844  BOUNDED wbdc 12872 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-bd0 12845  ax-bdan 12847  ax-bdsb 12854  ax-bdsep 12916 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rab 2400  df-v 2660  df-in 3045  df-ss 3052  df-bdc 12873 This theorem is referenced by:  bj-inex  12939
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