Is there a difference between $\forall x\forall x'$ and $\forall x, x'$ (in set theory more specifically)? And if there is what is it?
For example the definition of injective function:
Is this:
$$\forall x\forall x'(f(x) = f(x') \implies x=x')$$
the same as this:
$$\forall x, x'(x \ne x' \implies f(x) \ne f(x'))$$
In the concrete case you have given, it is actually the same since $x,x'$ belong to the same set, which is the domain of $f$. In general, I would say that the first one is used when there are different domains or sets (for example, $\forall x \in X, \forall y \in Y$), and the second one when the domain or set is the same (for example, $\forall x, y \in X$). I would be grateful if you upvoted this answer if it was helpful.