Theorem bd0r 16212

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 16211) 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 16211	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 105  BOUNDED wbd 16199

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 16200

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bdbi  16213  bdstab  16214  bddc  16215  bd3or  16216  bd3an  16217  bdfal  16220  bdxor  16223  bj-bdcel  16224  bdab  16225  bdcdeq  16226  bdne  16240  bdnel  16241  bdreu  16242  bdrmo  16243  bdsbcALT  16246  bdss  16251  bdeq0  16254  bdvsn  16261  bdop  16262  bdeqsuc  16268  bj-bdind  16317