![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > dfnot | GIF version |
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.) |
Ref | Expression |
---|---|
dfnot | ⊢ (¬ φ ↔ (φ → ⊥ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fal 1249 | . 2 ⊢ ¬ ⊥ | |
2 | mtt 609 | . 2 ⊢ (¬ ⊥ → (¬ φ ↔ (φ → ⊥ ))) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (¬ φ ↔ (φ → ⊥ )) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 ⊥ wfal 1247 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-fal 1248 |
This theorem is referenced by: inegd 1262 pclem6 1264 alnex 1385 alexim 1533 difin 3168 indifdir 3187 bj-axempty2 9349 |
Copyright terms: Public domain | W3C validator |