Theorem bd0r 15317

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 15316) 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 15316	1 |- BOUNDED ps

Syntax hints:   <-> wb 105  BOUNDED wbd 15304

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 15305

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bdbi  15318  bdstab  15319  bddc  15320  bd3or  15321  bd3an  15322  bdfal  15325  bdxor  15328  bj-bdcel  15329  bdab  15330  bdcdeq  15331  bdne  15345  bdnel  15346  bdreu  15347  bdrmo  15348  bdsbcALT  15351  bdss  15356  bdeq0  15359  bdvsn  15366  bdop  15367  bdeqsuc  15373  bj-bdind  15422