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Theorem bdceq 7288
Description: Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
Hypothesis
Ref Expression
bdceq.1 A = B
Assertion
Ref Expression
bdceq (BOUNDED ABOUNDED B)

Proof of Theorem bdceq
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 bdceq.1 . . . . 5 A = B
21eleq2i 2082 . . . 4 (x Ax B)
32bdeq 7269 . . 3 (BOUNDED x ABOUNDED x B)
43albii 1335 . 2 (xBOUNDED x AxBOUNDED x B)
5 df-bdc 7287 . 2 (BOUNDED AxBOUNDED x A)
6 df-bdc 7287 . 2 (BOUNDED BxBOUNDED x B)
74, 5, 63bitr4i 201 1 (BOUNDED ABOUNDED B)
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1224   = wceq 1226   wcel 1370  BOUNDED wbd 7258  BOUNDED wbdc 7286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1312  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-4 1377  ax-17 1396  ax-ial 1405  ax-ext 2000  ax-bd0 7259
This theorem depends on definitions:  df-bi 110  df-cleq 2011  df-clel 2014  df-bdc 7287
This theorem is referenced by:  bdceqi  7289
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