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Theorem bdccsb 11239
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 11225 . . . 4 BOUNDED 𝑧𝐴
32bdsbc 11237 . . 3 BOUNDED [𝑦 / 𝑥]𝑧𝐴
43bdcab 11228 . 2 BOUNDED {𝑧[𝑦 / 𝑥]𝑧𝐴}
5 df-csb 2923 . 2 𝑦 / 𝑥𝐴 = {𝑧[𝑦 / 𝑥]𝑧𝐴}
64, 5bdceqir 11223 1 BOUNDED 𝑦 / 𝑥𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1436  {cab 2071  [wsbc 2829  csb 2922  BOUNDED wbdc 11219
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 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470  ax-ext 2067  ax-bd0 11192  ax-bdsb 11201
This theorem depends on definitions:  df-bi 115  df-clab 2072  df-cleq 2078  df-clel 2081  df-sbc 2830  df-csb 2923  df-bdc 11220
This theorem is referenced by: (None)
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