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Theorem bdab 13835
Description: Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
Hypothesis
Ref Expression
bdab.1  |- BOUNDED  ph
Assertion
Ref Expression
bdab  |- BOUNDED  x  e.  { y  |  ph }

Proof of Theorem bdab
StepHypRef Expression
1 bdab.1 . . 3  |- BOUNDED  ph
21ax-bdsb 13819 . 2  |- BOUNDED  [ x  /  y ] ph
3 df-clab 2157 . 2  |-  ( x  e.  { y  | 
ph }  <->  [ x  /  y ] ph )
42, 3bd0r 13822 1  |- BOUNDED  x  e.  { y  |  ph }
Colors of variables: wff set class
Syntax hints:   [wsb 1755    e. wcel 2141   {cab 2156  BOUNDED wbd 13809
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-bd0 13810  ax-bdsb 13819
This theorem depends on definitions:  df-bi 116  df-clab 2157
This theorem is referenced by:  bdcab  13846  bdsbcALT  13856
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