Theorem bd0r 16420

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 16419) 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 16419	1 |- BOUNDED ps

Syntax hints:   <-> wb 105  BOUNDED wbd 16407

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 16408

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bdbi  16421  bdstab  16422  bddc  16423  bd3or  16424  bd3an  16425  bdfal  16428  bdxor  16431  bj-bdcel  16432  bdab  16433  bdcdeq  16434  bdne  16448  bdnel  16449  bdreu  16450  bdrmo  16451  bdsbcALT  16454  bdss  16459  bdeq0  16462  bdvsn  16469  bdop  16470  bdeqsuc  16476  bj-bdind  16525