Theorem bd0r 9945

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 9944) 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 123	. 2 ⊢ (𝜑 ↔ 𝜓)
4	1, 3	bd0 9944	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 98  BOUNDED wbd 9932

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-bd0 9933

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  bdbi  9946  bdstab  9947  bddc  9948  bd3or  9949  bd3an  9950  bdfal  9953  bdxor  9956  bj-bdcel  9957  bdab  9958  bdcdeq  9959  bdne  9973  bdnel  9974  bdreu  9975  bdrmo  9976  bdsbcALT  9979  bdss  9984  bdeq0  9987  bdvsn  9994  bdop  9995  bdeqsuc  10001  bj-bdind  10054