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

Theorem dfnot 1305
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 1294 . 2 ¬ ⊥
2 mtt 643 . 2 (¬ ⊥ → (¬ 𝜑 ↔ (𝜑 → ⊥)))
31, 2ax-mp 7 1 𝜑 ↔ (𝜑 → ⊥))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  wfal 1292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293
This theorem is referenced by:  inegd  1306  pclem6  1308  alnex  1431  alexim  1579  difin  3222  indifdir  3241  recvguniq  10269  bj-axempty2  11142
  Copyright terms: Public domain W3C validator