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Theorem bdccsb 13227
Description: A class resulting from proper substitution of a setvar for a setvar in a bounded class is bounded. (Contributed by BJ, 16-Oct-2019.)
Hypothesis
Ref Expression
bdccsb.1  |- BOUNDED  A
Assertion
Ref Expression
bdccsb  |- BOUNDED 
[_ y  /  x ]_ A

Proof of Theorem bdccsb
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 bdccsb.1 . . . . 5  |- BOUNDED  A
21bdeli 13213 . . . 4  |- BOUNDED  z  e.  A
32bdsbc 13225 . . 3  |- BOUNDED  [. y  /  x ]. z  e.  A
43bdcab 13216 . 2  |- BOUNDED  { z  |  [. y  /  x ]. z  e.  A }
5 df-csb 3007 . 2  |-  [_ y  /  x ]_ A  =  { z  |  [. y  /  x ]. z  e.  A }
64, 5bdceqir 13211 1  |- BOUNDED 
[_ y  /  x ]_ A
Colors of variables: wff set class
Syntax hints:    e. wcel 1481   {cab 2126   [.wsbc 2912   [_csb 3006  BOUNDED wbdc 13207
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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122  ax-bd0 13180  ax-bdsb 13189
This theorem depends on definitions:  df-bi 116  df-clab 2127  df-cleq 2133  df-clel 2136  df-sbc 2913  df-csb 3007  df-bdc 13208
This theorem is referenced by: (None)
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