Theorem bd0r 13782

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 13781) 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 13781	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 104  BOUNDED wbd 13769

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 13770

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  bdbi  13783  bdstab  13784  bddc  13785  bd3or  13786  bd3an  13787  bdfal  13790  bdxor  13793  bj-bdcel  13794  bdab  13795  bdcdeq  13796  bdne  13810  bdnel  13811  bdreu  13812  bdrmo  13813  bdsbcALT  13816  bdss  13821  bdeq0  13824  bdvsn  13831  bdop  13832  bdeqsuc  13838  bj-bdind  13887