Theorem bd0r 17056

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 17055) 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 17055	1 |- BOUNDED ps

This proof depends on syntax axioms:   <-> wb 105  BOUNDED wbd 17043

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-bd0 17044

This proof depends on definitions:  df-bi 117

This theorem is used by:  bdbi  17057  bdstab  17058  bddc  17059  bd3or  17060  bd3an  17061  bdfal  17064  bdxor  17067  bj-bdcel  17068  bdab  17069  bdcdeq  17070  bdne  17084  bdnel  17085  bdreu  17086  bdrmo  17087  bdsbcALT  17090  bdss  17095  bdeq0  17098  bdvsn  17105  bdop  17106  bdeqsuc  17112  bj-bdind  17161