Theorem bd0r 15557

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 15556) 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 15556	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 105  BOUNDED wbd 15544

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 15545

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bdbi  15558  bdstab  15559  bddc  15560  bd3or  15561  bd3an  15562  bdfal  15565  bdxor  15568  bj-bdcel  15569  bdab  15570  bdcdeq  15571  bdne  15585  bdnel  15586  bdreu  15587  bdrmo  15588  bdsbcALT  15591  bdss  15596  bdeq0  15599  bdvsn  15606  bdop  15607  bdeqsuc  15613  bj-bdind  15662