Theorem bd0r 13820

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 13819) 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 13819	1 |- BOUNDED ps

Syntax hints:   <-> wb 104  BOUNDED wbd 13807

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 13808

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  bdbi  13821  bdstab  13822  bddc  13823  bd3or  13824  bd3an  13825  bdfal  13828  bdxor  13831  bj-bdcel  13832  bdab  13833  bdcdeq  13834  bdne  13848  bdnel  13849  bdreu  13850  bdrmo  13851  bdsbcALT  13854  bdss  13859  bdeq0  13862  bdvsn  13869  bdop  13870  bdeqsuc  13876  bj-bdind  13925