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Theorem bdceq 14564
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 2244 . . . 4 (𝑥𝐴𝑥𝐵)
32bdeq 14545 . . 3 (BOUNDED 𝑥𝐴BOUNDED 𝑥𝐵)
43albii 1470 . 2 (∀𝑥BOUNDED 𝑥𝐴 ↔ ∀𝑥BOUNDED 𝑥𝐵)
5 df-bdc 14563 . 2 (BOUNDED 𝐴 ↔ ∀𝑥BOUNDED 𝑥𝐴)
6 df-bdc 14563 . 2 (BOUNDED 𝐵 ↔ ∀𝑥BOUNDED 𝑥𝐵)
74, 5, 63bitr4i 212 1 (BOUNDED 𝐴BOUNDED 𝐵)
Colors of variables: wff set class
Syntax hints:  wb 105  wal 1351   = wceq 1353  wcel 2148  BOUNDED wbd 14534  BOUNDED wbdc 14562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159  ax-bd0 14535
This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173  df-bdc 14563
This theorem is referenced by:  bdceqi  14565
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