Theorem imandc 894

Description: Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 692, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 25-Apr-2018.)

Assertion

Ref	Expression
imandc	|- (DECID ps -> ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) )

Proof of Theorem imandc

Step	Hyp	Ref	Expression
1		dcstab 849	. 2 |- (DECID ps -> STAB ps )
2		imanst 893	. 2 |- (STAB ps -> ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) )
3	1, 2	syl 14	1 |- (DECID ps -> ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  STAB wstab 835  DECID wdc 839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840

This theorem is referenced by:  annimdc  943  isprm3  12640