Theorem bdceq 9962

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 2104	. . . 4 |- ( x e. A <-> x e. B )
3	2	bdeq 9943	. . 3 |- (BOUNDED x e. A <-> BOUNDED x e. B )
4	3	albii 1359	. 2 |- ( A. xBOUNDED x e. A <-> A. xBOUNDED x e. B )
5		df-bdc 9961	. 2 |- (BOUNDED A <-> A. xBOUNDED x e. A )
6		df-bdc 9961	. 2 |- (BOUNDED B <-> A. xBOUNDED x e. B )
7	4, 5, 6	3bitr4i 201	1 |- (BOUNDED A <-> BOUNDED B )

Syntax hints:   <-> wb 98   A.wal 1241   = wceq 1243   e. wcel 1393  BOUNDED wbd 9932  BOUNDED wbdc 9960

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022  ax-bd0 9933

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036  df-bdc 9961

This theorem is referenced by:  bdceqi  9963