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Theorem bdrabexg 11680
Description: Bounded version of rabexg 3980. (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 3106 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
2 bdrabexg.bdc . . . 4 BOUNDED 𝐴
3 bdrabexg.bd . . . 4 BOUNDED 𝜑
42, 3bdcrab 11626 . . 3 BOUNDED {𝑥𝐴𝜑}
54bdssexg 11678 . 2 (({𝑥𝐴𝜑} ⊆ 𝐴𝐴𝑉) → {𝑥𝐴𝜑} ∈ V)
61, 5mpan 415 1 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1438  {crab 2363  Vcvv 2619  wss 2999  BOUNDED wbd 11586  BOUNDED wbdc 11614
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-bd0 11587  ax-bdan 11589  ax-bdsb 11596  ax-bdsep 11658
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-in 3005  df-ss 3012  df-bdc 11615
This theorem is referenced by:  bj-inex  11681
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