Theorem bd0r 14662

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 14661) 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 14661	1 |- BOUNDED ps

Syntax hints:   <-> wb 105  BOUNDED wbd 14649

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 14650

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bdbi  14663  bdstab  14664  bddc  14665  bd3or  14666  bd3an  14667  bdfal  14670  bdxor  14673  bj-bdcel  14674  bdab  14675  bdcdeq  14676  bdne  14690  bdnel  14691  bdreu  14692  bdrmo  14693  bdsbcALT  14696  bdss  14701  bdeq0  14704  bdvsn  14711  bdop  14712  bdeqsuc  14718  bj-bdind  14767