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Theorem bdph 9970
 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 9966 . . . 4 BOUNDED 𝑦 ∈ {𝑥𝜑}
3 df-clab 2027 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
42, 3bd0 9944 . . 3 BOUNDED [𝑦 / 𝑥]𝜑
54ax-bdsb 9942 . 2 BOUNDED [𝑥 / 𝑦][𝑦 / 𝑥]𝜑
6 sbid2v 1872 . 2 ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑𝜑)
75, 6bd0 9944 1 BOUNDED 𝜑
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1393  [wsb 1645  {cab 2026  BOUNDED wbd 9932  BOUNDED wbdc 9960 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-bd0 9933  ax-bdsb 9942 This theorem depends on definitions:  df-bi 110  df-sb 1646  df-clab 2027  df-bdc 9961 This theorem is referenced by:  bds  9971
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