Theorem notnot 630

Description: Double negation introduction. Theorem *2.12 of [WhiteheadRussell] p. 101. The converse need not hold. It holds exactly for stable propositions (by definition, see df-stab 833) and in particular for decidable propositions (see notnotrdc 845). See also notnotnot 635. (Contributed by NM, 28-Dec-1992.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)

Assertion

Ref	Expression
notnot	|- ( ph -> -. -. ph )

Proof of Theorem notnot

Step	Hyp	Ref	Expression
1		id 19	. 2 |- ( -. ph -> -. ph )
2	1	con2i 628	1 |- ( ph -> -. -. ph )

Syntax hints:   -. wn 3   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-in1 615  ax-in2 616

This theorem is referenced by:  notnotd  631  con3d  632  notnotnot  635  notnoti  646  pm3.24  695  biortn  747  dcn  844  con1dc  858  notnotbdc  874  imanst  890  eueq2dc  2953  ddifstab  3313  ifnotdc  3618  ismkvnex  7283  xrlttri3  9954  nltpnft  9971  ngtmnft  9974  bj-nnsn  15869  bj-nndcALT  15894  bdnthALT  15970