Theorem bd0r 15555

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 15554) biconditional in the hypothesis, to work better with definitions (𝜓 is the definiendum that one wants to prove bounded). (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bd0r.min	⊢ BOUNDED 𝜑
bd0r.maj	⊢ (𝜓 ↔ 𝜑)

Assertion

Ref	Expression
bd0r	⊢ BOUNDED 𝜓

Proof of Theorem bd0r

Step	Hyp	Ref	Expression
1		bd0r.min	. 2 ⊢ BOUNDED 𝜑
2		bd0r.maj	. . 3 ⊢ (𝜓 ↔ 𝜑)
3	2	bicomi 132	. 2 ⊢ (𝜑 ↔ 𝜓)
4	1, 3	bd0 15554	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 105  BOUNDED wbd 15542

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 15543

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bdbi  15556  bdstab  15557  bddc  15558  bd3or  15559  bd3an  15560  bdfal  15563  bdxor  15566  bj-bdcel  15567  bdab  15568  bdcdeq  15569  bdne  15583  bdnel  15584  bdreu  15585  bdrmo  15586  bdsbcALT  15589  bdss  15594  bdeq0  15597  bdvsn  15604  bdop  15605  bdeqsuc  15611  bj-bdind  15660