Theorem bd0r 15030

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 15029) 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 15029	1 |- BOUNDED ps

Syntax hints:   <-> wb 105  BOUNDED wbd 15017

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 15018

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bdbi  15031  bdstab  15032  bddc  15033  bd3or  15034  bd3an  15035  bdfal  15038  bdxor  15041  bj-bdcel  15042  bdab  15043  bdcdeq  15044  bdne  15058  bdnel  15059  bdreu  15060  bdrmo  15061  bdsbcALT  15064  bdss  15069  bdeq0  15072  bdvsn  15079  bdop  15080  bdeqsuc  15086  bj-bdind  15135