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Theorem bdph 10826
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 10822 . . . 4 BOUNDED 𝑦 ∈ {𝑥𝜑}
3 df-clab 2069 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
42, 3bd0 10800 . . 3 BOUNDED [𝑦 / 𝑥]𝜑
54ax-bdsb 10798 . 2 BOUNDED [𝑥 / 𝑦][𝑦 / 𝑥]𝜑
6 sbid2v 1914 . 2 ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑𝜑)
75, 6bd0 10800 1 BOUNDED 𝜑
Colors of variables: wff set class
Syntax hints:  wcel 1434  [wsb 1686  {cab 2068  BOUNDED wbd 10788  BOUNDED wbdc 10816
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-bd0 10789  ax-bdsb 10798
This theorem depends on definitions:  df-bi 115  df-sb 1687  df-clab 2069  df-bdc 10817
This theorem is referenced by:  bds  10827
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