Term for "existence" and uniqueness, when you already have solution candidate?

Question

For any given property $P(x)$ defined for $x\in X$, we can ask "does there exist an $x\in X$ such that $P(x)$?", and "if so, is this $x$ the unique element of $X$ that satisfies this".

This is called "existence and uniqueness".

However, assume now that the $x$ has already been identified as a candidate for having property $x$, and that we have to prove that $P(x)$ holds, and is unique.

What shorthand term would we use for "this particular $x$ has the property $P(x)$"?

Basically, I am looking for an equivalent version of the statement "prove existence and uniqueness" in the case where the candidate $x$ is already known: "prove ..... and uniqueness".

The reason I'm asking, is in order to shorten the phrase "prove that $x$ satisfies the property $P(x)$ and uniquely so".

EDIT: what I want to do is be able to write a proof like this:

First we prove [existence] (substitute existence for the term im looking for).

.....

Now we prove uniqueness.

....

Answer 1

Let object $x$ and property $P(\check{x})$ be given, where $\check{x}$ is a free variable.

If you want clarity, then you can go with

"Prove $P(x)$ holds and for each object $y$, if $P(y)$ holds, then $y=x$."

If you want brevity, then you can go with

"Prove $x$ is the unique object satisfying $P$."

Answer 2

If you have a specific $x$ in mind, then you are able to identify that $x$ because it is the unique element which satisfies some other formula / property $Q$. Then the sentence "For all $y$, $Q(y)$ iff $P(y)$" says that there is only one element which satisfies $P$, and that element is $x$.