Definition df-dc 836

Description: Propositions which are known to be true or false are called decidable. The (classical) Law of the Excluded Middle corresponds to the principle that all propositions are decidable, but even given intuitionistic logic, particular kinds of propositions may be decidable (for example, the proposition that two natural numbers are equal will be decidable under most sets of axioms).

Our notation for decidability is a connective DECID which we place before the formula in question. For example, DECID x = y corresponds to " x = y is decidable".

We could transform intuitionistic logic to classical logic by adding unconditional forms of condc 854, exmiddc 837, peircedc 915, or notnotrdc 844, any of which would correspond to the assertion that all propositions are decidable.

(Contributed by Jim Kingdon, 11-Mar-2018.)

Assertion

Ref	Expression
df-dc	|- (DECID ph <-> ( ph \/ -. ph ) )

Detailed syntax breakdown of Definition df-dc

Step	Hyp	Ref	Expression
1		wph	. . 3 wff ph
2	1	wdc 835	. 2 wff DECID ph
3	1	wn 3	. . 3 wff -. ph
4	1, 3	wo 709	. 2 wff ( ph \/ -. ph )
5	2, 4	wb 105	1 wff (DECID ph <-> ( ph \/ -. ph ) )

This definition is referenced by:  exmiddc  837  pm2.1dc  838  dcbid  839  dcim  842  dcn  843  notnotrdc  844  stdcndc  846  stdcndcOLD  847  stdcn  848  dcnnOLD  850  nndc  852  condcOLD  855  pm2.61ddc  862  pm5.18dc  884  pm2.13dc  886  pm2.25dc  894  pm2.85dc  906  pm5.12dc  911  pm5.14dc  912  pm5.55dc  914  peircedc  915  pm5.54dc  919  dcand  934  dcor  937  pm5.62dc  947  pm5.63dc  948  pm4.83dc  953  xordc1  1404  biassdc  1406  dcfromnotnotr  1458  dcfromcon  1459  dcfrompeirce  1460  19.30dc  1641  nfdc  1673  exmodc  2095  moexexdc  2129  dcne  2378  eueq2dc  2937  eueq3dc  2938  abvor0dc  3475  dcun  3561  ifcldcd  3598  ifnotdc  3599  ifandc  3600  ifmdc  3602  exmid01  4232  exmidsssnc  4237  exmidundif  4240  exmidundifim  4241  exmid1stab  4242  ontriexmidim  4559  ontri2orexmidim  4609  dcextest  4618  nndceq0  4655  nndceq  6566  nndcel  6567  fidceq  6939  tridc  6969  finexdc  6972  unsnfidcex  6990  unsnfidcel  6991  undifdcss  6993  dcfi  7056  ctssdccl  7186  nninfisollem0  7205  nninfisollemne  7206  nninfisollemeq  7207  nninfisol  7208  fodjuomnilemdc  7219  omniwomnimkv  7242  exmidonfinlem  7272  exmidfodomrlemr  7281  exmidfodomrlemrALT  7282  exmidaclem  7291  elni2  7398  indpi  7426  distrlem4prl  7668  distrlem4pru  7669  sup3exmid  9001  zdcle  9419  zdclt  9420  uzin  9651  elnn1uz2  9698  eluzdc  9701  xnn0dcle  9894  xrpnfdc  9934  xrmnfdc  9935  fztri3or  10131  fzdcel  10132  fzneuz  10193  exfzdc  10333  infssuzex  10340  qdclt  10352  fzfig  10539  sumdc  11540  zsumdc  11566  isum  11567  sum0  11570  fisumss  11574  isumss2  11575  fsumsplit  11589  sumsplitdc  11614  isumlessdc  11678  zproddc  11761  iprodap  11762  iprodap0  11764  prod0  11767  fprodssdc  11772  fprodsplitdc  11778  fprodsplit  11779  fprodunsn  11786  ef0lem  11842  nnwosdc  12231  prmdc  12323  pclemdc  12482  1arith  12561  ennnfonelemdc  12641  ctiunctlemudc  12679  bj-trdc  15482  bj-fadc  15484  bj-dcstab  15486  bj-nndcALT  15488  decidi  15525  decidr  15526  bj-charfunr  15540  bddc  15558  subctctexmid  15731  triap  15760  dceqnconst  15791  dcapnconst  15792  nconstwlpolem0  15794  nconstwlpo  15797  dftest  15806