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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 1250 | . 2 ⊢ ¬ ⊥ | |
2 | mtt 610 | . 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 1248 |
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 1246 df-fal 1249 |
This theorem is referenced by: inegd 1263 pclem6 1265 alnex 1388 alexim 1536 difin 3174 indifdir 3193 recvguniq 9593 bj-axempty2 10014 |
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