Theorem imanst 856

Description: Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2012.)

Assertion

Ref	Expression
imanst	|- (STAB ps -> ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) )

Proof of Theorem imanst

Step	Hyp	Ref	Expression
1		notnot 601	. . . 4 |- ( ps -> -. -. ps )
2		df-stab 799	. . . . 5 |- (STAB ps <-> ( -. -. ps -> ps ) )
3	2	biimpi 119	. . . 4 |- (STAB ps -> ( -. -. ps -> ps ) )
4	1, 3	impbid2 142	. . 3 |- (STAB ps -> ( ps <-> -. -. ps ) )
5	4	imbi2d 229	. 2 |- (STAB ps -> ( ( ph -> ps ) <-> ( ph -> -. -. ps ) ) )
6		imnan 662	. 2 |- ( ( ph -> -. -. ps ) <-> -. ( ph /\ -. ps ) )
7	5, 6	syl6bb 195	1 |- (STAB ps -> ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104  STAB wstab 798

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587

This theorem depends on definitions:  df-bi 116  df-stab 799

This theorem is referenced by:  imandc  857  dfss4st  3275