Theorem bdceq 13417

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.

Step	Hyp	Ref	Expression
1		bdceq.1	. . . . 5 |- A = B
2	1	eleq2i 2224	. . . 4 |- ( x e. A <-> x e. B )
3	2	bdeq 13398	. . 3 |- (BOUNDED x e. A <-> BOUNDED x e. B )
4	3	albii 1450	. 2 |- ( A. xBOUNDED x e. A <-> A. xBOUNDED x e. B )
5		df-bdc 13416	. 2 |- (BOUNDED A <-> A. xBOUNDED x e. A )
6		df-bdc 13416	. 2 |- (BOUNDED B <-> A. xBOUNDED x e. B )
7	4, 5, 6	3bitr4i 211	1 |- (BOUNDED A <-> BOUNDED B )

Syntax hints:   <-> wb 104   A.wal 1333   = wceq 1335   e. wcel 2128  BOUNDED wbd 13387  BOUNDED wbdc 13415

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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-ext 2139  ax-bd0 13388

This theorem depends on definitions:  df-bi 116  df-cleq 2150  df-clel 2153  df-bdc 13416

This theorem is referenced by:  bdceqi  13418