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.

Step	Hyp	Ref	Expression
1		bdceq.1	. . . . 5 |- A = B
2	1	eleq2i 2146	. . . 4 |- ( x e. A <-> x e. B )
3	2	bdeq 10799	. . 3 |- (BOUNDED x e. A <-> BOUNDED x e. B )
4	3	albii 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 )
7	4, 5, 6	3bitr4i 210	1 |- (BOUNDED A <-> BOUNDED B )

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