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Theorem bdccsb 11094
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 11080 . . . 4  |- BOUNDED  z  e.  A
32bdsbc 11092 . . 3  |- BOUNDED  [. y  /  x ]. z  e.  A
43bdcab 11083 . 2  |- BOUNDED  { z  |  [. y  /  x ]. z  e.  A }
5 df-csb 2920 . 2  |-  [_ y  /  x ]_ A  =  { z  |  [. y  /  x ]. z  e.  A }
64, 5bdceqir 11078 1  |- BOUNDED 
[_ y  /  x ]_ A
Colors of variables: wff set class
Syntax hints:    e. wcel 1434   {cab 2069   [.wsbc 2826   [_csb 2919  BOUNDED wbdc 11074
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-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2065  ax-bd0 11047  ax-bdsb 11056
This theorem depends on definitions:  df-bi 115  df-clab 2070  df-cleq 2076  df-clel 2079  df-sbc 2827  df-csb 2920  df-bdc 11075
This theorem is referenced by: (None)
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