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Theorem bds 14463
Description: Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 14434; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 14434. (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 14461 . . 3  |- BOUNDED  { x  |  ph }
3 bds.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
43cbvabv 2302 . . 3  |-  { x  |  ph }  =  {
y  |  ps }
52, 4bdceqi 14455 . 2  |- BOUNDED  { y  |  ps }
65bdph 14462 1  |- BOUNDED  ps
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   {cab 2163  BOUNDED wbd 14424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bd0 14425  ax-bdsb 14434
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-bdc 14453
This theorem is referenced by: (None)
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