Theorem exmid 894

Description: Law of excluded middle, also called the principle of tertium non datur. Theorem *2.11 of [WhiteheadRussell] p. 101. It says that something is either true or not true; there are no in-between values of truth. This is an essential distinction of our classical logic and is not a theorem of intuitionistic logic. In intuitionistic logic, if this statement is true for some 𝜑, then 𝜑 is decidable. (Contributed by NM, 29-Dec-1992.)

Assertion

Ref	Expression
exmid	⊢ (𝜑 ∨ ¬ 𝜑)

Proof of Theorem exmid

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (¬ 𝜑 → ¬ 𝜑)
2	1	orri 862	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ∨ wo 847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-or 848

This theorem is referenced by:  exmidd  895  pm5.62  1020  pm5.63  1021  pm4.83  1026  cases  1042  xpima  6135  fvresval  7299  ssfi  9097  ixxun  13282  trclfvg  14940  mreexexd  17572  lgsquadlem2  27308  numclwwlk3lem2  30346  elimifd  32505  elim2ifim  32507  iocinif  32737  hasheuni  34051  voliune  34195  volfiniune  34196  bnj1304  34785  wl-cases2-dnf  37485  cnambfre  37647  tsim1  38109  rp-isfinite6  43491  or3or  43996  uunT1  44753  onfrALTVD  44864  ax6e2ndeqVD  44882  ax6e2ndeqALT  44904  testable  49786