Theorem bd0r 10883

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 10882) 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 130	. 2 ⊢ (𝜑 ↔ 𝜓)
4	1, 3	bd0 10882	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 103  BOUNDED wbd 10870

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-bd0 10871

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  bdbi  10884  bdstab  10885  bddc  10886  bd3or  10887  bd3an  10888  bdfal  10891  bdxor  10894  bj-bdcel  10895  bdab  10896  bdcdeq  10897  bdne  10911  bdnel  10912  bdreu  10913  bdrmo  10914  bdsbcALT  10917  bdss  10922  bdeq0  10925  bdvsn  10932  bdop  10933  bdeqsuc  10939  bj-bdind  10992