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Theorem bdph 12882
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 12878 . . . 4 BOUNDED 𝑦 ∈ {𝑥𝜑}
3 df-clab 2102 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
42, 3bd0 12856 . . 3 BOUNDED [𝑦 / 𝑥]𝜑
54ax-bdsb 12854 . 2 BOUNDED [𝑥 / 𝑦][𝑦 / 𝑥]𝜑
6 sbid2v 1947 . 2 ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑𝜑)
75, 6bd0 12856 1 BOUNDED 𝜑
Colors of variables: wff set class
Syntax hints:  wcel 1463  [wsb 1718  {cab 2101  BOUNDED wbd 12844  BOUNDED wbdc 12872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-bd0 12845  ax-bdsb 12854
This theorem depends on definitions:  df-bi 116  df-sb 1719  df-clab 2102  df-bdc 12873
This theorem is referenced by:  bds  12883
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