Theorem bd0r 17073

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 17072) 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 17072	1 ⊢ BOUNDED 𝜓

This proof depends on syntax axioms:   ↔ wb 105  BOUNDED wbd 17060

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-bd0 17061

This proof depends on definitions:  df-bi 117

This theorem is used by:  bdbi  17074  bdstab  17075  bddc  17076  bd3or  17077  bd3an  17078  bdfal  17081  bdxor  17084  bj-bdcel  17085  bdab  17086  bdcdeq  17087  bdne  17101  bdnel  17102  bdreu  17103  bdrmo  17104  bdsbcALT  17107  bdss  17112  bdeq0  17115  bdvsn  17122  bdop  17123  bdeqsuc  17129  bj-bdind  17178