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Theorem bj-bdcel 13483
Description: Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)
Hypothesis
Ref Expression
bj-bdcel.bd BOUNDED 𝑦 = 𝐴
Assertion
Ref Expression
bj-bdcel BOUNDED 𝐴𝑥
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem bj-bdcel
StepHypRef Expression
1 bj-bdcel.bd . . 3 BOUNDED 𝑦 = 𝐴
21ax-bdex 13465 . 2 BOUNDED𝑦𝑥 𝑦 = 𝐴
3 risset 2485 . 2 (𝐴𝑥 ↔ ∃𝑦𝑥 𝑦 = 𝐴)
42, 3bd0r 13471 1 BOUNDED 𝐴𝑥
Colors of variables: wff set class
Syntax hints:   = wceq 1335  wcel 2128  wrex 2436  BOUNDED wbd 13458
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-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-ial 1514  ax-bd0 13459  ax-bdex 13465
This theorem depends on definitions:  df-bi 116  df-clel 2153  df-rex 2441
This theorem is referenced by:  bj-bd0el  13514  bj-bdsucel  13528
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