Verify if the following expression is true

Question

If: $$\exists! x P(x) \iff \exists x\forall y (P(y) \leftrightarrow y=x)$$ Then: $$\begin{split}\neg(\exists! x P(x)) \iff \neg(\exists x\forall y (P(y) \leftrightarrow y=x))\\ \iff (\neg\exists x)(\neg\forall y)\neg (P(y) \leftrightarrow y=x)\\ \iff \forall x \exists y \space \neg (P(y) \leftrightarrow y=x)\\ \iff \forall x \exists y (P(y) \oplus (y=x))\end{split}$$

Answer 1

There was a misstep on the second line. $~\neg (\exists x~\psi) \Leftrightarrow (\neg\exists x)~\psi~$ but $~\forall y~\phi \nLeftrightarrow \neg \forall y~\neg\phi~$.

Instead you should have used the duality rules. $~\neg\exists x~\phi\Leftrightarrow \forall x~\neg\phi~$. Then $~\neg\forall y~\psi\Leftrightarrow \exists y~\neg\psi$ will give the third line.

$$\begin{split}\neg(\exists! x P(x)) &\iff \neg(\exists x\forall y (P(y) \leftrightarrow y=x))\\ &\iff \require{cancel}\cancel{(\neg\exists x)(\neg\forall y)\neg (P(y) \leftrightarrow y=x)} & \quad&\forall x~\neg\forall y~(P(y)\leftrightarrow y=x)\\ &\iff \forall x \exists y \space \neg (P(y) \leftrightarrow y=x)\\ &\iff \forall x \exists y (P(y) \oplus (y=x))\end{split}$$

However, you did reach the correct conclusion in the end.