Sometimes I see expressions like $\forall a,b$ which is a quantifier followed by two variables. Does $\forall a,b\in X: P(a,b)$ mean $\forall a: \forall b\in X: P(a,b)$? So is $\forall a,b$ just an abbreviation?
In words: Is "for all a and b we have P(a,b)" just an abbreviation for "for all a: for all b: P(a,b)"?
It just means "for any two elements of X, P is true about them."
I believe that what it reduces to in terms of the actual formal language grammar (or at least the one I use) is: $\forall x \ni (\exists y \ni (x\in X\land y\in X), P(x,y)),(\textrm{insert generic predicate here})$.
The problem is that it's less intuitive when written out in those terms, even though it ultimately means "given x and y in X, ( P(x,y) ) is true.
I would NOT recommend trying to think about it in terms of its official version, though. There's nothing to be gained in doing so; and the only thing lost is a ton of intuition, which is our guiding light as mathematicians.