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Theorem bdph 11387
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 11383 . . . 4 BOUNDED 𝑦 ∈ {𝑥𝜑}
3 df-clab 2075 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
42, 3bd0 11361 . . 3 BOUNDED [𝑦 / 𝑥]𝜑
54ax-bdsb 11359 . 2 BOUNDED [𝑥 / 𝑦][𝑦 / 𝑥]𝜑
6 sbid2v 1920 . 2 ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑𝜑)
75, 6bd0 11361 1 BOUNDED 𝜑
Colors of variables: wff set class
Syntax hints:  wcel 1438  [wsb 1692  {cab 2074  BOUNDED wbd 11349  BOUNDED wbdc 11377
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-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-bd0 11350  ax-bdsb 11359
This theorem depends on definitions:  df-bi 115  df-sb 1693  df-clab 2075  df-bdc 11378
This theorem is referenced by:  bds  11388
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