Theorem bddc 13026

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

Hypothesis

Ref	Expression
bdstab.1	|- BOUNDED ph

Assertion

Ref	Expression
bddc	|- BOUNDED DECID ph

Proof of Theorem bddc

Step	Hyp	Ref	Expression
1		bdstab.1	. . 3 |- BOUNDED ph
2	1	ax-bdn 13015	. . 3 |- BOUNDED -. ph
3	1, 2	ax-bdor 13014	. 2 |- BOUNDED ( ph \/ -. ph )
4		df-dc 820	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
5	3, 4	bd0r 13023	1 |- BOUNDED DECID ph

Syntax hints:   -. wn 3   \/ wo 697  DECID wdc 819  BOUNDED wbd 13010

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 13011  ax-bdor 13014  ax-bdn 13015

This theorem depends on definitions:  df-bi 116  df-dc 820

This theorem is referenced by: (None)