Theorem bd0r 16851

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 16850) 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 16850	1 |- BOUNDED ps

This proof depends on syntax axioms:   <-> wb 105  BOUNDED wbd 16838

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 16839

This proof depends on definitions:  df-bi 117

This theorem is used by:  bdbi  16852  bdstab  16853  bddc  16854  bd3or  16855  bd3an  16856  bdfal  16859  bdxor  16862  bj-bdcel  16863  bdab  16864  bdcdeq  16865  bdne  16879  bdnel  16880  bdreu  16881  bdrmo  16882  bdsbcALT  16885  bdss  16890  bdeq0  16893  bdvsn  16900  bdop  16901  bdeqsuc  16907  bj-bdind  16956