How to find that $\exists !xP(x) \equiv \exists x(P(x) \land \forall y(P(y) \rightarrow y=x))$ when only LHS is given? $$\exists !xP(x) \equiv \exists x(P(x) \land \forall y(P(y) \rightarrow y=x))$$
By writing down what each of the side means in English sentence, I can show that both of them are are the same thing.
LHS: There exists unique value of x such that it satisfies P(x)
RHS: There exists some values x which satisfy P(x) and if any other value also satisfies P, then it is equal to x. This essentially means that all x are the same value. So it is unique.
But I'm unable to figure out a way of going to the RHS when only the LHS is given. How do I find the RHS when only the LHS is given?
 A: Yes, you have done it.
You have taken the LHS. $~\exists! x~P(x)$
You have interpreted this to be equivalent to the statement, "There exists unique value of $x$ such that it satisfies $P(x)$".
You have established that, "There exists some values $x$ which satisfy $P(x)$ and/but if any value satisfies $P$, then it is equal to $x$," has an equivalent sematic meaning.
Then you translated that semantics into the syntax: $\exists x~(P(x)\wedge \forall y~(P(y)\to y=x))$
Then by the chain rule for equivalence, you are done. $$\exists!x~P(x)~\iff~\exists x~(P(x)\wedge \forall y~(P(y)\to y=x))$$
That is all that is required.   It is very much the reason why the $\exists!$ quantifier is defined that way.
A: If as you say the LHS is defined as the RHS, then note that the proof that they are equivalent is immediate, because the meaning of a definition is that the notation being defined (in this case the LHS) is equivalent to something (in this case the RHS). So there is really nothing to prove.
Note that one is free to define all sorts of notation, but they may not be useful. What makes this particular notation involving "$\exists!$" defined this way useful is that it correctly captures the intended notion of uniqueness. Beware that this justification of usefulness cannot be proven, since there is no way we can explain "uniqueness" without already understanding it or defining it in the way it is done here!
