Theorem bddc 16866

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 16855	. . 3 ⊢ BOUNDED ¬ 𝜑
3	1, 2	ax-bdor 16854	. 2 ⊢ BOUNDED (𝜑 ∨ ¬ 𝜑)
4		df-dc 847	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
5	3, 4	bd0r 16863	1 ⊢ BOUNDED DECID 𝜑

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 720  DECID wdc 846  BOUNDED wbd 16850

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-bd0 16851  ax-bdor 16854  ax-bdn 16855

This proof depends on definitions:  df-bi 117  df-dc 847

This theorem is used by: (None)