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I have been trying to solve the question as mentioned above and I have failed to so do. Please provide me a solution.

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    $\begingroup$ What have you tried? What logical rules are you allowed to use? $\endgroup$ – ZeroXLR Apr 8 at 9:28
  • $\begingroup$ See the answer to similar post. $\endgroup$ – Mauro ALLEGRANZA Apr 8 at 9:33
  • $\begingroup$ @ZeroXLR This is my 1st post on logic. I am new in logic. I would like to request you to use very basic logical rules and please solve. $\endgroup$ – user1942348 Apr 8 at 9:34
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    $\begingroup$ Please, provide details about the proof system you are asked to use : Natural Deduction ? $\endgroup$ – Mauro ALLEGRANZA Apr 8 at 9:49
  • $\begingroup$ Natural Deduction $\endgroup$ – user1942348 Apr 8 at 9:55
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First of all for a formula $\varphi(x)$ we have

$\exists x\varphi(x) \iff \bigvee_a{\varphi(a)}$

and

$\forall x\varphi(x)\iff\bigwedge_a{\varphi(a)}$

In your case

$\neg\exists x\varphi(x) \iff \neg\bigvee_a{\varphi(a)} \iff \bigwedge_a{\neg\varphi(a)} \iff \forall x\neg\varphi(x)$

The above follows from the de morgan rules

Finally negating again gives

$\exists x\varphi(x) \iff \neg\neg\exists x\varphi(x) \iff \neg\forall x\neg\varphi(x)$

You need to make it a bit more formal but thats the idea

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