Theorem bd0r 16721

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 16720) biconditional in the hypothesis, to work better with definitions ( ps is the definiendum that one wants to prove bounded). (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bd0r.min	|- BOUNDED ph
bd0r.maj	|- ( ps <-> ph )

Assertion

Ref	Expression
bd0r	|- BOUNDED ps

Proof of Theorem bd0r

Step	Hyp	Ref	Expression
1		bd0r.min	. 2 |- BOUNDED ph
2		bd0r.maj	. . 3 |- ( ps <-> ph )
3	2	bicomi 132	. 2 |- ( ph <-> ps )
4	1, 3	bd0 16720	1 |- BOUNDED ps

Syntax hints:   <-> wb 105  BOUNDED wbd 16708

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-bd0 16709

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bdbi  16722  bdstab  16723  bddc  16724  bd3or  16725  bd3an  16726  bdfal  16729  bdxor  16732  bj-bdcel  16733  bdab  16734  bdcdeq  16735  bdne  16749  bdnel  16750  bdreu  16751  bdrmo  16752  bdsbcALT  16755  bdss  16760  bdeq0  16763  bdvsn  16770  bdop  16771  bdeqsuc  16777  bj-bdind  16826