Theorem bdceq 15488

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 2263	. . . 4 |- ( x e. A <-> x e. B )
3	2	bdeq 15469	. . 3 |- (BOUNDED x e. A <-> BOUNDED x e. B )
4	3	albii 1484	. 2 |- ( A. xBOUNDED x e. A <-> A. xBOUNDED x e. B )
5		df-bdc 15487	. 2 |- (BOUNDED A <-> A. xBOUNDED x e. A )
6		df-bdc 15487	. 2 |- (BOUNDED B <-> A. xBOUNDED x e. B )
7	4, 5, 6	3bitr4i 212	1 |- (BOUNDED A <-> BOUNDED B )

Syntax hints:   <-> wb 105   A.wal 1362   = wceq 1364   e. wcel 2167  BOUNDED wbd 15458  BOUNDED wbdc 15486

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178  ax-bd0 15459

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192  df-bdc 15487

This theorem is referenced by:  bdceqi  15489