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Theorem bdph 15040
Description: A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.)
Hypothesis
Ref Expression
bdph.1 BOUNDED {𝑥𝜑}
Assertion
Ref Expression
bdph BOUNDED 𝜑

Proof of Theorem bdph
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 bdph.1 . . . . 5 BOUNDED {𝑥𝜑}
21bdeli 15036 . . . 4 BOUNDED 𝑦 ∈ {𝑥𝜑}
3 df-clab 2176 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
42, 3bd0 15014 . . 3 BOUNDED [𝑦 / 𝑥]𝜑
54ax-bdsb 15012 . 2 BOUNDED [𝑥 / 𝑦][𝑦 / 𝑥]𝜑
6 sbid2v 2008 . 2 ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑𝜑)
75, 6bd0 15014 1 BOUNDED 𝜑
Colors of variables: wff set class
Syntax hints:  [wsb 1773  wcel 2160  {cab 2175  BOUNDED wbd 15002  BOUNDED wbdc 15030
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-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-bd0 15003  ax-bdsb 15012
This theorem depends on definitions:  df-bi 117  df-sb 1774  df-clab 2176  df-bdc 15031
This theorem is referenced by:  bds  15041
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