Theorem bd0r 15960

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 15959) 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 15959	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 105  BOUNDED wbd 15947

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 15948

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bdbi  15961  bdstab  15962  bddc  15963  bd3or  15964  bd3an  15965  bdfal  15968  bdxor  15971  bj-bdcel  15972  bdab  15973  bdcdeq  15974  bdne  15988  bdnel  15989  bdreu  15990  bdrmo  15991  bdsbcALT  15994  bdss  15999  bdeq0  16002  bdvsn  16009  bdop  16010  bdeqsuc  16016  bj-bdind  16065