Theorem bd0r 10908

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 10907) 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 130	. 2 |- ( ph <-> ps )
4	1, 3	bd0 10907	1 |- BOUNDED ps

Syntax hints:   <-> wb 103  BOUNDED wbd 10895

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-bd0 10896

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  bdbi  10909  bdstab  10910  bddc  10911  bd3or  10912  bd3an  10913  bdfal  10916  bdxor  10919  bj-bdcel  10920  bdab  10921  bdcdeq  10922  bdne  10936  bdnel  10937  bdreu  10938  bdrmo  10939  bdsbcALT  10942  bdss  10947  bdeq0  10950  bdvsn  10957  bdop  10958  bdeqsuc  10964  bj-bdind  11017