Theorem bd0r 16297

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 16296) 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 16296	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 105  BOUNDED wbd 16284

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 16285

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bdbi  16298  bdstab  16299  bddc  16300  bd3or  16301  bd3an  16302  bdfal  16305  bdxor  16308  bj-bdcel  16309  bdab  16310  bdcdeq  16311  bdne  16325  bdnel  16326  bdreu  16327  bdrmo  16328  bdsbcALT  16331  bdss  16336  bdeq0  16339  bdvsn  16346  bdop  16347  bdeqsuc  16353  bj-bdind  16402