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Theorem bdph 13153
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 13149 . . . 4  |- BOUNDED  y  e.  { x  |  ph }
3 df-clab 2126 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
42, 3bd0 13127 . . 3  |- BOUNDED  [ y  /  x ] ph
54ax-bdsb 13125 . 2  |- BOUNDED  [ x  /  y ] [ y  /  x ] ph
6 sbid2v 1971 . 2  |-  ( [ x  /  y ] [ y  /  x ] ph  <->  ph )
75, 6bd0 13127 1  |- BOUNDED  ph
Colors of variables: wff set class
Syntax hints:    e. wcel 1480   [wsb 1735   {cab 2125  BOUNDED wbd 13115  BOUNDED wbdc 13143
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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-bd0 13116  ax-bdsb 13125
This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2126  df-bdc 13144
This theorem is referenced by:  bds  13154
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