# “Prove that for any proposition $Ax$, $(∃x)(Ax) \iff \lnot(∀x)(\lnot Ax)$”

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

• What have you tried? What logical rules are you allowed to use? – ZeroXLR Apr 8 at 9:28
• See the answer to similar post. – Mauro ALLEGRANZA Apr 8 at 9:33
• @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. – user1942348 Apr 8 at 9:34
• Please, provide details about the proof system you are asked to use : Natural Deduction ? – Mauro ALLEGRANZA Apr 8 at 9:49
• Natural Deduction – user1942348 Apr 8 at 9:55

First of all for a formula $$\varphi(x)$$ we have

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

and

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

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