Is this natural deduction proof of $\exists x \neg Px \vdash \neg \forall x Px$ correct?

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When it comes to proofs there is no way to tell whether I have done correct or not. In the solution they did in another way which makes me wonder if this correct? For future question, how can I verify whether it is correct or not? I have heard of websites that can check but can't find any that works. enter image description here