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I want to prove $\vdash \neg \neg A \leftrightarrow A$ without using $RAA$ and $\bot$ rules. the part that $\vdash A \to \neg \neg A$ is simple but I can't prove the other part. is there any possibility for proving $\vdash \neg \neg A \to A$ without using $RAA$ and $\bot$ rule in natural deduction?

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    $\begingroup$ You cannot prove the "other part" because it is not intuitionistically valid. $\endgroup$ – Mauro ALLEGRANZA Nov 28 '16 at 14:26
  • $\begingroup$ @MauroALLEGRANZA thank you $\endgroup$ – Karo Nov 28 '16 at 14:28
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    $\begingroup$ See Intuitionistic Logic for the basic facts about it. $\endgroup$ – Mauro ALLEGRANZA Nov 28 '16 at 14:29

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