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 832) and in particular for decidable propositions (see notnotrdc 844). 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  694  biortn  746  dcn  843  con1dc  857  notnotbdc  873  imanst  889  eueq2dc  2933  ddifstab  3291  ifnotdc  3594  ismkvnex  7214  xrlttri3  9863  nltpnft  9880  ngtmnft  9883  bj-nnsn  15225  bj-nndcALT  15250  bdnthALT  15327