How To prove the logical equivalence for uniqueness quantifier

Question

Could you please help me out with this little issue, I need to know how to prove the following assertion. I'll be very thankful for your efforts :) Thanks in advance,

$$\exists !xP(x) \equiv \exists x(P(x) \land \forall y(P(y) \rightarrow y=x))$$

Answer 1

$$\exists !xP(x) \equiv \exists x(P(x) \land \forall y(P(y) \rightarrow y=x))$$

The right hand side of the equivalence is taken to be the definition of uniqueness operator, where $P(x)$ is any predicate about $x$.

Both sides of the equivalence can be translated to

"There exists a unique $x$ such that $P(x)$" $\;\equiv \;\;$"There exists one and only one $x$ such that $P(x)$".

The right hand side simply defines what it means to assert the existence of one and only one $x$ such that $P(x)$:

$(1)$ there exists $x$ such that $P(x)$ (existence of $x$ for which $P(x)$)

and that

$(2)$ for any $y$, if $P(y)$ is true, then $y$ must be (must equal) $x$ (uniqueness of $x$ such that $P(x)$).

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