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Theorem bdccsb 9953
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 9939 . . . 4  |- BOUNDED  z  e.  A
32bdsbc 9951 . . 3  |- BOUNDED  [. y  /  x ]. z  e.  A
43bdcab 9942 . 2  |- BOUNDED  { z  |  [. y  /  x ]. z  e.  A }
5 df-csb 2853 . 2  |-  [_ y  /  x ]_ A  =  { z  |  [. y  /  x ]. z  e.  A }
64, 5bdceqir 9937 1  |- BOUNDED 
[_ y  /  x ]_ A
Colors of variables: wff set class
Syntax hints:    e. wcel 1393   {cab 2026   [.wsbc 2764   [_csb 2852  BOUNDED wbdc 9933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022  ax-bd0 9906  ax-bdsb 9915
This theorem depends on definitions:  df-bi 110  df-clab 2027  df-cleq 2033  df-clel 2036  df-sbc 2765  df-csb 2853  df-bdc 9934
This theorem is referenced by: (None)
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