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Theorem bdccsb 12750
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 12736 . . . 4  |- BOUNDED  z  e.  A
32bdsbc 12748 . . 3  |- BOUNDED  [. y  /  x ]. z  e.  A
43bdcab 12739 . 2  |- BOUNDED  { z  |  [. y  /  x ]. z  e.  A }
5 df-csb 2972 . 2  |-  [_ y  /  x ]_ A  =  { z  |  [. y  /  x ]. z  e.  A }
64, 5bdceqir 12734 1  |- BOUNDED 
[_ y  /  x ]_ A
Colors of variables: wff set class
Syntax hints:    e. wcel 1463   {cab 2101   [.wsbc 2878   [_csb 2971  BOUNDED wbdc 12730
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497  ax-ext 2097  ax-bd0 12703  ax-bdsb 12712
This theorem depends on definitions:  df-bi 116  df-clab 2102  df-cleq 2108  df-clel 2111  df-sbc 2879  df-csb 2972  df-bdc 12731
This theorem is referenced by: (None)
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