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Theorem bj-bdcel 15047
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 15029 . 2 BOUNDED𝑦𝑥 𝑦 = 𝐴
3 risset 2518 . 2 (𝐴𝑥 ↔ ∃𝑦𝑥 𝑦 = 𝐴)
42, 3bd0r 15035 1 BOUNDED 𝐴𝑥
Colors of variables: wff set class
Syntax hints:   = wceq 1364  wcel 2160  wrex 2469  BOUNDED wbd 15022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545  ax-bd0 15023  ax-bdex 15029
This theorem depends on definitions:  df-bi 117  df-clel 2185  df-rex 2474
This theorem is referenced by:  bj-bd0el  15078  bj-bdsucel  15092
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