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Theorem bds 11099
Description: Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 11070; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 11070. (Contributed by BJ, 19-Nov-2019.)
Hypotheses
Ref Expression
bds.bd  |- BOUNDED  ph
bds.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
bds  |- BOUNDED  ps
Distinct variable groups:    ps, x    ph, y
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem bds
StepHypRef Expression
1 bds.bd . . . 4  |- BOUNDED  ph
21bdcab 11097 . . 3  |- BOUNDED  { x  |  ph }
3 bds.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
43cbvabv 2208 . . 3  |-  { x  |  ph }  =  {
y  |  ps }
52, 4bdceqi 11091 . 2  |- BOUNDED  { y  |  ps }
65bdph 11098 1  |- BOUNDED  ps
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   {cab 2071  BOUNDED wbd 11060
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-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-bd0 11061  ax-bdsb 11070
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-bdc 11089
This theorem is referenced by: (None)
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