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Theorem bdph 10908
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 10904 . . . 4  |- BOUNDED  y  e.  { x  |  ph }
3 df-clab 2070 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
42, 3bd0 10882 . . 3  |- BOUNDED  [ y  /  x ] ph
54ax-bdsb 10880 . 2  |- BOUNDED  [ x  /  y ] [ y  /  x ] ph
6 sbid2v 1915 . 2  |-  ( [ x  /  y ] [ y  /  x ] ph  <->  ph )
75, 6bd0 10882 1  |- BOUNDED  ph
Colors of variables: wff set class
Syntax hints:    e. wcel 1434   [wsb 1687   {cab 2069  BOUNDED wbd 10870  BOUNDED wbdc 10898
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 10871  ax-bdsb 10880
This theorem depends on definitions:  df-bi 115  df-sb 1688  df-clab 2070  df-bdc 10899
This theorem is referenced by:  bds  10909
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