Theorem bd0r 13860

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 13859) 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 131	. 2 ⊢ (𝜑 ↔ 𝜓)
4	1, 3	bd0 13859	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 104  BOUNDED wbd 13847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-bd0 13848

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  bdbi  13861  bdstab  13862  bddc  13863  bd3or  13864  bd3an  13865  bdfal  13868  bdxor  13871  bj-bdcel  13872  bdab  13873  bdcdeq  13874  bdne  13888  bdnel  13889  bdreu  13890  bdrmo  13891  bdsbcALT  13894  bdss  13899  bdeq0  13902  bdvsn  13909  bdop  13910  bdeqsuc  13916  bj-bdind  13965