Theorem bd0r 16970

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 16969) 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 16969	1 ⊢ BOUNDED 𝜓

This proof depends on syntax axioms:   ↔ wb 105  BOUNDED wbd 16957

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 16958

This proof depends on definitions:  df-bi 117

This theorem is used by:  bdbi  16971  bdstab  16972  bddc  16973  bd3or  16974  bd3an  16975  bdfal  16978  bdxor  16981  bj-bdcel  16982  bdab  16983  bdcdeq  16984  bdne  16998  bdnel  16999  bdreu  17000  bdrmo  17001  bdsbcALT  17004  bdss  17009  bdeq0  17012  bdvsn  17019  bdop  17020  bdeqsuc  17026  bj-bdind  17075