Theorem dcstab 839

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

Assertion

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

Proof of Theorem dcstab

Step	Hyp	Ref	Expression
1		notnotrdc 838	. 2 |- (DECID ph -> ( -. -. ph -> ph ) )
2		df-stab 826	. 2 |- (STAB ph <-> ( -. -. ph -> ph ) )
3	1, 2	sylibr 133	1 |- (DECID ph -> STAB ph )

Syntax hints:   -. wn 3   -> wi 4  STAB wstab 825  DECID wdc 829

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 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830

This theorem is referenced by:  stdcndc  840  stdcndcOLD  841  condc  848  imandc  884  sbthlemi3  6936