Theorem bd0r 15471

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 15470) 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 15470	1 |- BOUNDED ps

Syntax hints:   <-> wb 105  BOUNDED wbd 15458

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 15459

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bdbi  15472  bdstab  15473  bddc  15474  bd3or  15475  bd3an  15476  bdfal  15479  bdxor  15482  bj-bdcel  15483  bdab  15484  bdcdeq  15485  bdne  15499  bdnel  15500  bdreu  15501  bdrmo  15502  bdsbcALT  15505  bdss  15510  bdeq0  15513  bdvsn  15520  bdop  15521  bdeqsuc  15527  bj-bdind  15576