Theorem bd0r 17068

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 17067) 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 17067	1 ⊢ BOUNDED 𝜓

This proof depends on syntax axioms:   ↔ wb 105  BOUNDED wbd 17055

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 17056

This proof depends on definitions:  df-bi 117

This theorem is used by:  bdbi  17069  bdstab  17070  bddc  17071  bd3or  17072  bd3an  17073  bdfal  17076  bdxor  17079  bj-bdcel  17080  bdab  17081  bdcdeq  17082  bdne  17096  bdnel  17097  bdreu  17098  bdrmo  17099  bdsbcALT  17102  bdss  17107  bdeq0  17110  bdvsn  17117  bdop  17118  bdeqsuc  17124  bj-bdind  17173