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Theorem bdph 15055
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  { x  |  ph }
Assertion
Ref Expression
bdph  |- BOUNDED  ph

Proof of Theorem bdph
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 bdph.1 . . . . 5  |- BOUNDED  { x  |  ph }
21bdeli 15051 . . . 4  |- BOUNDED  y  e.  { x  |  ph }
3 df-clab 2176 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
42, 3bd0 15029 . . 3  |- BOUNDED  [ y  /  x ] ph
54ax-bdsb 15027 . 2  |- BOUNDED  [ x  /  y ] [ y  /  x ] ph
6 sbid2v 2008 . 2  |-  ( [ x  /  y ] [ y  /  x ] ph  <->  ph )
75, 6bd0 15029 1  |- BOUNDED  ph
Colors of variables: wff set class
Syntax hints:   [wsb 1773    e. wcel 2160   {cab 2175  BOUNDED wbd 15017  BOUNDED wbdc 15045
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 15018  ax-bdsb 15027
This theorem depends on definitions:  df-bi 117  df-sb 1774  df-clab 2176  df-bdc 15046
This theorem is referenced by:  bds  15056
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