Theorem bd0r 16949

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 16948) 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 16948	1 |- BOUNDED ps

This proof depends on syntax axioms:   <-> wb 105  BOUNDED wbd 16936

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 16937

This proof depends on definitions:  df-bi 117

This theorem is used by:  bdbi  16950  bdstab  16951  bddc  16952  bd3or  16953  bd3an  16954  bdfal  16957  bdxor  16960  bj-bdcel  16961  bdab  16962  bdcdeq  16963  bdne  16977  bdnel  16978  bdreu  16979  bdrmo  16980  bdsbcALT  16983  bdss  16988  bdeq0  16991  bdvsn  16998  bdop  16999  bdeqsuc  17005  bj-bdind  17054