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Theorem bdccsb 10809
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 10795 . . . 4 BOUNDED 𝑧𝐴
32bdsbc 10807 . . 3 BOUNDED [𝑦 / 𝑥]𝑧𝐴
43bdcab 10798 . 2 BOUNDED {𝑧[𝑦 / 𝑥]𝑧𝐴}
5 df-csb 2910 . 2 𝑦 / 𝑥𝐴 = {𝑧[𝑦 / 𝑥]𝑧𝐴}
64, 5bdceqir 10793 1 BOUNDED 𝑦 / 𝑥𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1434  {cab 2068  [wsbc 2816  csb 2909  BOUNDED wbdc 10789
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 2064  ax-bd0 10762  ax-bdsb 10771
This theorem depends on definitions:  df-bi 115  df-clab 2069  df-cleq 2075  df-clel 2078  df-sbc 2817  df-csb 2910  df-bdc 10790
This theorem is referenced by: (None)
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