Theorem dcimpstab 790

Description: Decidability implies stability. The converse is not necessarily true. (Contributed by David A. Wheeler, 13-Aug-2018.)

Assertion

Ref	Expression
dcimpstab	|- (DECID ph -> STAB ph )

Proof of Theorem dcimpstab

Step	Hyp	Ref	Expression
1		notnotrdc 789	. 2 |- (DECID ph -> ( -. -. ph -> ph ) )
2		df-stab 776	. 2 |- (STAB ph <-> ( -. -. ph -> ph ) )
3	1, 2	sylibr 132	1 |- (DECID ph -> STAB ph )

Syntax hints:   -. wn 3   -> wi 4  STAB wstab 775  DECID wdc 780

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 580  ax-io 665

This theorem depends on definitions:  df-bi 115  df-stab 776  df-dc 781

This theorem is referenced by:  stabtestimpdc  862  sbthlemi3  6668