Theorem bd0r 13012

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 13011) 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 13011	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 104  BOUNDED wbd 12999

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 13000

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  bdbi  13013  bdstab  13014  bddc  13015  bd3or  13016  bd3an  13017  bdfal  13020  bdxor  13023  bj-bdcel  13024  bdab  13025  bdcdeq  13026  bdne  13040  bdnel  13041  bdreu  13042  bdrmo  13043  bdsbcALT  13046  bdss  13051  bdeq0  13054  bdvsn  13061  bdop  13062  bdeqsuc  13068  bj-bdind  13117