Theorem bddc 13710

Description: Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)

Hypothesis

Ref	Expression
bdstab.1	⊢ BOUNDED 𝜑

Assertion

Ref	Expression
bddc	⊢ BOUNDED DECID 𝜑

Proof of Theorem bddc

Step	Hyp	Ref	Expression
1		bdstab.1	. . 3 ⊢ BOUNDED 𝜑
2	1	ax-bdn 13699	. . 3 ⊢ BOUNDED ¬ 𝜑
3	1, 2	ax-bdor 13698	. 2 ⊢ BOUNDED (𝜑 ∨ ¬ 𝜑)
4		df-dc 825	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
5	3, 4	bd0r 13707	1 ⊢ BOUNDED DECID 𝜑

Syntax hints:  ¬ wn 3   ∨ wo 698  DECID wdc 824  BOUNDED wbd 13694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-bd0 13695  ax-bdor 13698  ax-bdn 13699

This theorem depends on definitions:  df-bi 116  df-dc 825

This theorem is referenced by: (None)