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Theorem bdph 13845
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 13841 . . . 4  |- BOUNDED  y  e.  { x  |  ph }
3 df-clab 2157 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
42, 3bd0 13819 . . 3  |- BOUNDED  [ y  /  x ] ph
54ax-bdsb 13817 . 2  |- BOUNDED  [ x  /  y ] [ y  /  x ] ph
6 sbid2v 1989 . 2  |-  ( [ x  /  y ] [ y  /  x ] ph  <->  ph )
75, 6bd0 13819 1  |- BOUNDED  ph
Colors of variables: wff set class
Syntax hints:   [wsb 1755    e. wcel 2141   {cab 2156  BOUNDED wbd 13807  BOUNDED wbdc 13835
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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-bd0 13808  ax-bdsb 13817
This theorem depends on definitions:  df-bi 116  df-sb 1756  df-clab 2157  df-bdc 13836
This theorem is referenced by:  bds  13846
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