Theorem bddc 13863

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 13852	. . 3 ⊢ BOUNDED ¬ 𝜑
3	1, 2	ax-bdor 13851	. 2 ⊢ BOUNDED (𝜑 ∨ ¬ 𝜑)
4		df-dc 830	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
5	3, 4	bd0r 13860	1 ⊢ BOUNDED DECID 𝜑

Syntax hints:  ¬ wn 3   ∨ wo 703  DECID wdc 829  BOUNDED wbd 13847

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 13848  ax-bdor 13851  ax-bdn 13852

This theorem depends on definitions:  df-bi 116  df-dc 830

This theorem is referenced by: (None)