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Theorem bds 12976
Description: Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 12947; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 12947. (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 12974 . . 3  |- BOUNDED  { x  |  ph }
3 bds.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
43cbvabv 2241 . . 3  |-  { x  |  ph }  =  {
y  |  ps }
52, 4bdceqi 12968 . 2  |- BOUNDED  { y  |  ps }
65bdph 12975 1  |- BOUNDED  ps
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   {cab 2103  BOUNDED wbd 12937
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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-bd0 12938  ax-bdsb 12947
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-bdc 12966
This theorem is referenced by: (None)
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