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Theorem bdceq 9277
 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 2101 . . . 4 (x Ax B)
32bdeq 9258 . . 3 (BOUNDED x ABOUNDED x B)
43albii 1356 . 2 (xBOUNDED x AxBOUNDED x B)
5 df-bdc 9276 . 2 (BOUNDED AxBOUNDED x A)
6 df-bdc 9276 . 2 (BOUNDED BxBOUNDED x B)
74, 5, 63bitr4i 201 1 (BOUNDED ABOUNDED B)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98  ∀wal 1240   = wceq 1242   ∈ wcel 1390  BOUNDED wbd 9247  BOUNDED wbdc 9275 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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019  ax-bd0 9248 This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-clel 2033  df-bdc 9276 This theorem is referenced by:  bdceqi  9278
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