Theorem bdeq 9943

Description: Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)

Hypothesis

Ref	Expression
bdeq.1	|- ( ph <-> ps )

Assertion

Ref	Expression
bdeq	|- (BOUNDED ph <-> BOUNDED ps )

Proof of Theorem bdeq

Step	Hyp	Ref	Expression
1		bdeq.1	. . 3 |- ( ph <-> ps )
2	1	ax-bd0 9933	. 2 |- (BOUNDED ph -> BOUNDED ps )
3	1	bicomi 123	. . 3 |- ( ps <-> ph )
4	3	ax-bd0 9933	. 2 |- (BOUNDED ps -> BOUNDED ph )
5	2, 4	impbii 117	1 |- (BOUNDED ph <-> BOUNDED ps )

Syntax hints:   <-> wb 98  BOUNDED wbd 9932

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-bd0 9933

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  bdceq  9962