Theorem bd0r 11671

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 11670) 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 11670	1 |- BOUNDED ps

Syntax hints:   <-> wb 103  BOUNDED wbd 11658

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 11659

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  bdbi  11672  bdstab  11673  bddc  11674  bd3or  11675  bd3an  11676  bdfal  11679  bdxor  11682  bj-bdcel  11683  bdab  11684  bdcdeq  11685  bdne  11699  bdnel  11700  bdreu  11701  bdrmo  11702  bdsbcALT  11705  bdss  11710  bdeq0  11713  bdvsn  11720  bdop  11721  bdeqsuc  11727  bj-bdind  11780