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Theorem bdccsb 13895
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 13881 . . . 4 BOUNDED 𝑧𝐴
32bdsbc 13893 . . 3 BOUNDED [𝑦 / 𝑥]𝑧𝐴
43bdcab 13884 . 2 BOUNDED {𝑧[𝑦 / 𝑥]𝑧𝐴}
5 df-csb 3050 . 2 𝑦 / 𝑥𝐴 = {𝑧[𝑦 / 𝑥]𝑧𝐴}
64, 5bdceqir 13879 1 BOUNDED 𝑦 / 𝑥𝐴
Colors of variables: wff set class
Syntax hints:  wcel 2141  {cab 2156  [wsbc 2955  csb 3049  BOUNDED wbdc 13875
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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152  ax-bd0 13848  ax-bdsb 13857
This theorem depends on definitions:  df-bi 116  df-clab 2157  df-cleq 2163  df-clel 2166  df-sbc 2956  df-csb 3050  df-bdc 13876
This theorem is referenced by: (None)
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