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Theorem bdph 11186
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 11182 . . . 4  |- BOUNDED  y  e.  { x  |  ph }
3 df-clab 2072 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
42, 3bd0 11160 . . 3  |- BOUNDED  [ y  /  x ] ph
54ax-bdsb 11158 . 2  |- BOUNDED  [ x  /  y ] [ y  /  x ] ph
6 sbid2v 1917 . 2  |-  ( [ x  /  y ] [ y  /  x ] ph  <->  ph )
75, 6bd0 11160 1  |- BOUNDED  ph
Colors of variables: wff set class
Syntax hints:    e. wcel 1436   [wsb 1689   {cab 2071  BOUNDED wbd 11148  BOUNDED wbdc 11176
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 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-bd0 11149  ax-bdsb 11158
This theorem depends on definitions:  df-bi 115  df-sb 1690  df-clab 2072  df-bdc 11177
This theorem is referenced by:  bds  11187
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