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Mirrors > Home > ILE Home > Th. List > nndc | GIF version |
Description: Double negation of decidability of a formula. Intuitionistic logic refutes undecidability (but does not prove decidability) of any formula. (Contributed by BJ, 9-Oct-2019.) |
Ref | Expression |
---|---|
nndc | ⊢ ¬ ¬ DECID 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnexmid 820 | . 2 ⊢ ¬ ¬ (𝜑 ∨ ¬ 𝜑) | |
2 | df-dc 805 | . . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
3 | 2 | notbii 642 | . 2 ⊢ (¬ DECID 𝜑 ↔ ¬ (𝜑 ∨ ¬ 𝜑)) |
4 | 1, 3 | mtbir 645 | 1 ⊢ ¬ ¬ DECID 𝜑 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∨ wo 682 DECID wdc 804 |
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 588 ax-in2 589 ax-io 683 |
This theorem depends on definitions: df-bi 116 df-dc 805 |
This theorem is referenced by: bj-nnst 12891 bj-dcdc 12892 bj-stdc 12893 |
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