Theorem bd0r 14580

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 14579) 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 14579	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 105  BOUNDED wbd 14567

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 14568

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bdbi  14581  bdstab  14582  bddc  14583  bd3or  14584  bd3an  14585  bdfal  14588  bdxor  14591  bj-bdcel  14592  bdab  14593  bdcdeq  14594  bdne  14608  bdnel  14609  bdreu  14610  bdrmo  14611  bdsbcALT  14614  bdss  14619  bdeq0  14622  bdvsn  14629  bdop  14630  bdeqsuc  14636  bj-bdind  14685