Theorem bj-dcstab 12966

Description: A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-dcstab	|- (DECID ph -> STAB ph )

Proof of Theorem bj-dcstab

Step	Hyp	Ref	Expression
1		df-dc 820	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		bj-trst 12956	. . 3 |- ( ph -> STAB ph )
3		bj-fast 12957	. . 3 |- ( -. ph -> STAB ph )
4	2, 3	jaoi 705	. 2 |- ( ( ph \/ -. ph ) -> STAB ph )
5	1, 4	sylbi 120	1 |- (DECID ph -> STAB ph )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 697  STAB wstab 815  DECID wdc 819

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-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820

This theorem is referenced by:  bj-nnbidc  12967