Theorem bd0r 17030

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 17029) 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 17029	1 ⊢ BOUNDED 𝜓

This proof depends on syntax axioms:   ↔ wb 105  BOUNDED wbd 17017

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 17018

This proof depends on definitions:  df-bi 117

This theorem is used by:  bdbi  17031  bdstab  17032  bddc  17033  bd3or  17034  bd3an  17035  bdfal  17038  bdxor  17041  bj-bdcel  17042  bdab  17043  bdcdeq  17044  bdne  17058  bdnel  17059  bdreu  17060  bdrmo  17061  bdsbcALT  17064  bdss  17069  bdeq0  17072  bdvsn  17079  bdop  17080  bdeqsuc  17086  bj-bdind  17135