Theorem bd0r 10616

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 10615) 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 130	. 2 ⊢ (𝜑 ↔ 𝜓)
4	1, 3	bd0 10615	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 103  BOUNDED wbd 10603

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-bd0 10604

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  bdbi  10617  bdstab  10618  bddc  10619  bd3or  10620  bd3an  10621  bdfal  10624  bdxor  10627  bj-bdcel  10628  bdab  10629  bdcdeq  10630  bdne  10644  bdnel  10645  bdreu  10646  bdrmo  10647  bdsbcALT  10650  bdss  10655  bdeq0  10658  bdvsn  10665  bdop  10666  bdeqsuc  10672  bj-bdind  10725