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Mirrors > Home > ILE Home > Th. List > notnotbdc | GIF version |
Description: Double negation equivalence for a decidable proposition. Like Theorem *4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent. The forward direction, notnot 619, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 13-Mar-2018.) |
Ref | Expression |
---|---|
notnotbdc | ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notnot 619 | . 2 ⊢ (𝜑 → ¬ ¬ 𝜑) | |
2 | notnotrdc 829 | . 2 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → 𝜑)) | |
3 | 1, 2 | impbid2 142 | 1 ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 DECID wdc 820 |
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 604 ax-in2 605 ax-io 699 |
This theorem depends on definitions: df-bi 116 df-dc 821 |
This theorem is referenced by: con1biidc 863 imordc 883 dfbi3dc 1379 alexdc 1599 |
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