$\underset{x}{\bigvee} \mathfrak{P}x \wedge \underset{x}{\bigwedge} \underset{y}{\bigwedge}(\mathfrak{P}x \wedge \mathfrak{P}y\rightarrow x=y)?$ I'm reading Behnke's fundamentals of mathematics, he written that the following proposition:
$$\underset{x}{\bigvee} \mathfrak{P}x \wedge \underset{x}{\bigwedge} \underset{y}{\bigwedge}(\mathfrak{P}x \wedge \mathfrak{P}y\rightarrow x=y)$$
Means that there exists only one even prime number. But I'm unable to see this assumption in the given symbols. $\mathfrak{P}$ stands for the predicate "is an even prime number". I also fail to see why not use for example: $\exists ! x$ such that $\mathfrak{P}x$ is true.
 A: The $\bigvee$ and $\bigwedge$ notations are old (I was baffled at the first time I saw them), but when you think about them they make some sense. When we say $\exists x\varphi(x)$ is true, say over the natural numbers, we really say that $\varphi(0)\lor\varphi(1)\lor\ldots$ is true, i.e. there is some $n$ such that $\varphi(n)$ holds. Similarly $\forall x$ behaves like a gigantic conjuction. And these two translate well to large universes too by the same idea.
So what does the statement says? First it says that there is in fact an even prime number. Then it says that if $x$ and $y$ are both even prime number, then $x=y$. Therefore there exists exactly one even prime number. This could also been written as:$$\bigvee_x\left(\mathfrak Px\land\bigwedge_y\Big(\mathfrak P(y)\rightarrow x=y\Big)\right)$$
Where this formulation says that there is an even prime number $x$, and every other even prime number is equal to $x$. So it has to be unique.
One reason not to use $\exists!x$ is that $\exists!x$ obscures a longer, and fully expressible, formula. So in the context of logic we sometimes prefer to write the longer formula. It also could be the case that this notation did not exist (or wasn't fully standard) when the text was written, which would make an excellent excuse!
A: The symbols translate to "There is an even prime number, and if x and y are even primes, they're identical". The second clause rules out there being two distinct even primes, and the first rules out the possibility that there are no even primes. It's basically the same thing as $\exists !$; the most straightforward way to define $\exists !x\phi(x)$ is $\exists x( \phi(x)\wedge \forall y(\phi(y)\to x=y))$.
