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Mirrors > Home > ILE Home > Th. List > Mathboxes > nnexmid | GIF version |
Description: Double negation of excluded middle. Intuitionistic logic refutes the negation of excluded middle (but, of course, does not prove excluded middle) for any formula. (Contributed by BJ, 9-Oct-2019.) |
Ref | Expression |
---|---|
nnexmid | ⊢ ¬ ¬ (𝜑 ∨ ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.24 668 | . 2 ⊢ ¬ (¬ 𝜑 ∧ ¬ ¬ 𝜑) | |
2 | ioran 710 | . 2 ⊢ (¬ (𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑 ∧ ¬ ¬ 𝜑)) | |
3 | 1, 2 | mtbir 637 | 1 ⊢ ¬ ¬ (𝜑 ∨ ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ∨ wo 670 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: nndc 12549 |
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