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Theorem bdceq 12842
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  A  <-> BOUNDED  B )

Proof of Theorem bdceq
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 bdceq.1 . . . . 5  |-  A  =  B
21eleq2i 2182 . . . 4  |-  ( x  e.  A  <->  x  e.  B )
32bdeq 12823 . . 3  |-  (BOUNDED  x  e.  A  <-> BOUNDED  x  e.  B )
43albii 1429 . 2  |-  ( A. xBOUNDED  x  e.  A  <->  A. xBOUNDED  x  e.  B )
5 df-bdc 12841 . 2  |-  (BOUNDED  A  <->  A. xBOUNDED  x  e.  A )
6 df-bdc 12841 . 2  |-  (BOUNDED  B  <->  A. xBOUNDED  x  e.  B )
74, 5, 63bitr4i 211 1  |-  (BOUNDED  A  <-> BOUNDED  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   A.wal 1312    = wceq 1314    e. wcel 1463  BOUNDED wbd 12812  BOUNDED wbdc 12840
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 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497  ax-ext 2097  ax-bd0 12813
This theorem depends on definitions:  df-bi 116  df-cleq 2108  df-clel 2111  df-bdc 12841
This theorem is referenced by:  bdceqi  12843
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