Theorem bd0r 17022

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 17021) 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 17021	1 ⊢ BOUNDED 𝜓

This proof depends on syntax axioms:   ↔ wb 105  BOUNDED wbd 17009

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 17010

This proof depends on definitions:  df-bi 117

This theorem is used by:  bdbi  17023  bdstab  17024  bddc  17025  bd3or  17026  bd3an  17027  bdfal  17030  bdxor  17033  bj-bdcel  17034  bdab  17035  bdcdeq  17036  bdne  17050  bdnel  17051  bdreu  17052  bdrmo  17053  bdsbcALT  17056  bdss  17061  bdeq0  17064  bdvsn  17071  bdop  17072  bdeqsuc  17078  bj-bdind  17127