Theorem bd0r 17017

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 17016) 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 17016	1 |- BOUNDED ps

This proof depends on syntax axioms:   <-> wb 105  BOUNDED wbd 17004

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 17005

This proof depends on definitions:  df-bi 117

This theorem is used by:  bdbi  17018  bdstab  17019  bddc  17020  bd3or  17021  bd3an  17022  bdfal  17025  bdxor  17028  bj-bdcel  17029  bdab  17030  bdcdeq  17031  bdne  17045  bdnel  17046  bdreu  17047  bdrmo  17048  bdsbcALT  17051  bdss  17056  bdeq0  17059  bdvsn  17066  bdop  17067  bdeqsuc  17073  bj-bdind  17122