Theorem bd0r 16644

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 16643) 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 16643	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 105  BOUNDED wbd 16631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-bd0 16632

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bdbi  16645  bdstab  16646  bddc  16647  bd3or  16648  bd3an  16649  bdfal  16652  bdxor  16655  bj-bdcel  16656  bdab  16657  bdcdeq  16658  bdne  16672  bdnel  16673  bdreu  16674  bdrmo  16675  bdsbcALT  16678  bdss  16683  bdeq0  16686  bdvsn  16693  bdop  16694  bdeqsuc  16700  bj-bdind  16749