For all unique notation? Is there notation for "for all unique..."? For instance, suppose you wanted to consider all distinct $x,y$ in some set $S$. Would we type $\forall !x,y\in S$?
Can we use "distinct" and "unique" interchangeably in this sense?
I've never seen notation for this and a google search/SE search did not find anything.
 A: The word "unique" means having a property that distinguishes it from all other things. In mathematics, this is meaningless if not followed by "such that" and the precise property that distinguishes it from everything else. Every $x$ is the unique $y$ such that $y=x$, so being unique without qualification is void of meaning.
The one useful use of "unique" in mathematics is therefore in the context "the is a unique $x$ (in a given set $X$) such that the property $P(x)$ holds", which translated into a logical formula gives $\exists x\in X:\bigl(P(x)\land \forall y\in X:P(y)\to x=y\bigr)$.
The word "distinct" applies to a a family of variables (often just two or three): the variables $x_i$ for $i$ in some index set $I$ are all distinct whenever $x_i=x_j$ implies $i=j$ (the map $I\to X:i\mapsto x_i$ is injective). It is not attached to universal quantification (saying "for all distinct $x$" is void of meaning), though the variables $x_i$ might have been introduced by quantification (either universal or existential, both are quite reasonable).
So there is nothing that links "unique" with "distinct", except that both are non-properties: they are void of meaning when applied (without qualification) to a single variable. I think what may be confusing you is the horrible abuse of "unique" that is sometimes used in combinatorics (or probability): determine the number of unique configurations of some kind. I never know what "unique" is supposed to mean in this context. It might mean "distinct" in the sense that each configuration is counted only once, but that is rather silly since this is implicit in counting things correctly. Sometimes "number of distinct" such and such may be used to indicate not attaching a multiplicity which the reader might otherwise be inclined to do: the number of distinct roots of a polynomial, or eigenvalues of a matrix. However one would never say "number of unique" in this context. More likely the "number of unique configurations" means count only classes for some (not explicitly indicated) equivalence relation of configurations, such as orbits for a symmetry group acting on the set of configurations (seatings around a round table...).
In any case "for all unique $x,y,z$" is meaningless. One could say "for all distinct $x,y,z$", which is a slightly informal way of saying for all $x,y,z$ such that $x\neq y\neq z\neq x$ (don't forget the last inequality!).
A: Here is a definition of "unique" that I got by Googling:

being the only one of its kind; unlike anything else

Let $X$ be a set. Let $x$ be an element of $X$. Is $x$ unique? Of course, in an utterly useless sort of way:  $x$ is unique because anything other than $x$ isn't $x$. I see the only interpretation of 

For all unique $x,y\in X$, ...

as being

For all $x,y\in X$ such that $x$ and $y$ are unique, ...

or in other words

For all $x,y\in X$, ...

Now, if you want to use the word "unique" to mean something other than what everyone else uses it to mean, e.g. "distinct", you're welcome to do so, but you should expect there to be confusion.
A: I'd be wary of using $\forall !$ as your notation for this. (See here for example.) The word 'unique' definitely doesn't mean what you want  to say, though. 'Unique' means 'one', and 'distinct' means... exactly the opposite!
You'd be better off writing "for all distinct $x,y \in S$".
Working formally, if you were trying to show that a property $\phi(x,y)$ holds for distinct $x,y \in S$ you could write
$$(\forall x,y \in S)(x \ne y \to \phi(x,y))$$
...but outside of logic there isn't really much need for this kind of notation.
