Theorem imandc 874

Description: Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 677, 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 829	. 2 |- (DECID ps -> STAB ps )
2		imanst 873	. 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 103   <-> wb 104  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-in1 603  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:  annimdc  921  isprm3  11788