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Theorem bdceq 10818
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 2146 . . . 4  |-  ( x  e.  A  <->  x  e.  B )
32bdeq 10799 . . 3  |-  (BOUNDED  x  e.  A  <-> BOUNDED  x  e.  B )
43albii 1400 . 2  |-  ( A. xBOUNDED  x  e.  A  <->  A. xBOUNDED  x  e.  B )
5 df-bdc 10817 . 2  |-  (BOUNDED  A  <->  A. xBOUNDED  x  e.  A )
6 df-bdc 10817 . 2  |-  (BOUNDED  B  <->  A. xBOUNDED  x  e.  B )
74, 5, 63bitr4i 210 1  |-  (BOUNDED  A  <-> BOUNDED  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   A.wal 1283    = wceq 1285    e. wcel 1434  BOUNDED wbd 10788  BOUNDED wbdc 10816
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 10789
This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-clel 2078  df-bdc 10817
This theorem is referenced by:  bdceqi  10819
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