Theorem bd0r 13707

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 13706) 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 13706	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 104  BOUNDED wbd 13694

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 13695

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  bdbi  13708  bdstab  13709  bddc  13710  bd3or  13711  bd3an  13712  bdfal  13715  bdxor  13718  bj-bdcel  13719  bdab  13720  bdcdeq  13721  bdne  13735  bdnel  13736  bdreu  13737  bdrmo  13738  bdsbcALT  13741  bdss  13746  bdeq0  13749  bdvsn  13756  bdop  13757  bdeqsuc  13763  bj-bdind  13812