Theorem bd0 10615

Description: A formula equivalent to a bounded one is bounded. See also bd0r 10616. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bd0.min	⊢ BOUNDED 𝜑
bd0.maj	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
bd0	⊢ BOUNDED 𝜓

Proof of Theorem bd0

Step	Hyp	Ref	Expression
1		bd0.min	. 2 ⊢ BOUNDED 𝜑
2		bd0.maj	. . 3 ⊢ (𝜑 ↔ 𝜓)
3	2	ax-bd0 10604	. 2 ⊢ (BOUNDED 𝜑 → BOUNDED 𝜓)
4	1, 3	ax-mp 7	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 103  BOUNDED wbd 10603

This theorem was proved from axioms:  ax-mp 7  ax-bd0 10604

This theorem is referenced by:  bd0r  10616  bdth  10622  bdnth  10625  bdnthALT  10626  bdph  10641  bdsbc  10649  bdsnss  10664  bdcint  10668  bdeqsuc  10672  bdcriota  10674  bj-axun2  10706