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Mirrors > Home > ILE Home > Th. List > Mathboxes > nndc | GIF version |
Description: Double negation of decidability of a formula. Intuitionistic logic refutes undecidability (but, of course, does not prove decidability) of any formula. (Contributed by BJ, 9-Oct-2019.) |
Ref | Expression |
---|---|
nndc | ⊢ ¬ ¬ DECID 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnexmid 12377 | . 2 ⊢ ¬ ¬ (𝜑 ∨ ¬ 𝜑) | |
2 | df-dc 784 | . . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
3 | 2 | notbii 632 | . 2 ⊢ (¬ DECID 𝜑 ↔ ¬ (𝜑 ∨ ¬ 𝜑)) |
4 | 1, 3 | mtbir 634 | 1 ⊢ ¬ ¬ DECID 𝜑 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∨ wo 667 DECID wdc 783 |
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 582 ax-in2 583 ax-io 668 |
This theorem depends on definitions: df-bi 116 df-dc 784 |
This theorem is referenced by: dcdc 12379 |
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