Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfnot GIF version

Theorem dfnot 1350
 Description: Given falsum, we can define the negation of a wff 𝜑 as the statement that a contradiction follows from assuming 𝜑. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
Assertion
Ref Expression
dfnot 𝜑 ↔ (𝜑 → ⊥))

Proof of Theorem dfnot
StepHypRef Expression
1 fal 1339 . 2 ¬ ⊥
2 mtt 675 . 2 (¬ ⊥ → (¬ 𝜑 ↔ (𝜑 → ⊥)))
31, 2ax-mp 5 1 𝜑 ↔ (𝜑 → ⊥))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  ⊥wfal 1337 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338 This theorem is referenced by:  inegd  1351  pclem6  1353  alnex  1476  alexim  1625  difin  3317  indifdir  3336  recvguniq  10798  logbgcd1irr  13090  bj-axempty2  13261
 Copyright terms: Public domain W3C validator