Theorem bd0r 13667

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 13666) 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 131	. 2 |- ( ph <-> ps )
4	1, 3	bd0 13666	1 |- BOUNDED ps

Syntax hints:   <-> wb 104  BOUNDED wbd 13654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-bd0 13655

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  bdbi  13668  bdstab  13669  bddc  13670  bd3or  13671  bd3an  13672  bdfal  13675  bdxor  13678  bj-bdcel  13679  bdab  13680  bdcdeq  13681  bdne  13695  bdnel  13696  bdreu  13697  bdrmo  13698  bdsbcALT  13701  bdss  13706  bdeq0  13709  bdvsn  13716  bdop  13717  bdeqsuc  13723  bj-bdind  13772