Theorem bd0r 15387

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 15386) 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 15386	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 105  BOUNDED wbd 15374

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 15375

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bdbi  15388  bdstab  15389  bddc  15390  bd3or  15391  bd3an  15392  bdfal  15395  bdxor  15398  bj-bdcel  15399  bdab  15400  bdcdeq  15401  bdne  15415  bdnel  15416  bdreu  15417  bdrmo  15418  bdsbcALT  15421  bdss  15426  bdeq0  15429  bdvsn  15436  bdop  15437  bdeqsuc  15443  bj-bdind  15492