Theorem bd0r 14547

Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 14546) 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 14546	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 105  BOUNDED wbd 14534

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 14535

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bdbi  14548  bdstab  14549  bddc  14550  bd3or  14551  bd3an  14552  bdfal  14555  bdxor  14558  bj-bdcel  14559  bdab  14560  bdcdeq  14561  bdne  14575  bdnel  14576  bdreu  14577  bdrmo  14578  bdsbcALT  14581  bdss  14586  bdeq0  14589  bdvsn  14596  bdop  14597  bdeqsuc  14603  bj-bdind  14652