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Theorem bdccsb 13047
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 / 𝑥𝐴

Proof of Theorem bdccsb
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bdccsb.1 . . . . 5 BOUNDED 𝐴
21bdeli 13033 . . . 4 BOUNDED 𝑧𝐴
32bdsbc 13045 . . 3 BOUNDED [𝑦 / 𝑥]𝑧𝐴
43bdcab 13036 . 2 BOUNDED {𝑧[𝑦 / 𝑥]𝑧𝐴}
5 df-csb 2999 . 2 𝑦 / 𝑥𝐴 = {𝑧[𝑦 / 𝑥]𝑧𝐴}
64, 5bdceqir 13031 1 BOUNDED 𝑦 / 𝑥𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1480  {cab 2123  [wsbc 2904  csb 2998  BOUNDED wbdc 13027
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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119  ax-bd0 13000  ax-bdsb 13009
This theorem depends on definitions:  df-bi 116  df-clab 2124  df-cleq 2130  df-clel 2133  df-sbc 2905  df-csb 2999  df-bdc 13028
This theorem is referenced by: (None)
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