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Mirrors > Home > MPE Home > Th. List > dfnot | Structured version Visualization version GIF version |
Description: Given falsum ⊥, we can define the negation of a wff 𝜑 as the statement that ⊥ 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 1556 | . 2 ⊢ ¬ ⊥ | |
2 | mtt 368 | . 2 ⊢ (¬ ⊥ → (¬ 𝜑 ↔ (𝜑 → ⊥))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (¬ 𝜑 ↔ (𝜑 → ⊥)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ⊥wfal 1554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-tru 1545 df-fal 1555 |
This theorem is referenced by: inegd 1562 ralf0 4397 bj-godellob 34414 irrdifflemf 35105 |
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