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 1338 | . 2 ⊢ ¬ ⊥ | |
2 | mtt 674 | . 2 ⊢ (¬ ⊥ → (¬ 𝜑 ↔ (𝜑 → ⊥))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (¬ 𝜑 ↔ (𝜑 → ⊥)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ⊥wfal 1336 |
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 603 ax-in2 604 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 |
This theorem is referenced by: inegd 1350 pclem6 1352 alnex 1475 alexim 1624 difin 3308 indifdir 3327 recvguniq 10760 bj-axempty2 13081 |
Copyright terms: Public domain | W3C validator |