Theorem bd0r 16863

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 16862) 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 16862	1 ⊢ BOUNDED 𝜓

This proof depends on syntax axioms:   ↔ wb 105  BOUNDED wbd 16850

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 16851

This proof depends on definitions:  df-bi 117

This theorem is used by:  bdbi  16864  bdstab  16865  bddc  16866  bd3or  16867  bd3an  16868  bdfal  16871  bdxor  16874  bj-bdcel  16875  bdab  16876  bdcdeq  16877  bdne  16891  bdnel  16892  bdreu  16893  bdrmo  16894  bdsbcALT  16897  bdss  16902  bdeq0  16905  bdvsn  16912  bdop  16913  bdeqsuc  16919  bj-bdind  16968