Here is my proof attempt:
Observe that $\forall{a\in{A},\forall{b\in{B}}} \equiv \forall{a,b}(a\in{A} \wedge b\in{B})$ and $\forall{(a,b)\in{A\times{B}}} \equiv \forall{a,b}((a,b)\in{A\times{B}})$. We show that $\forall{a,b}(a\in{A} \wedge b\in{B} \Leftrightarrow (a,b)\in{A\times{B}})$. Let $a\in{A} \wedge b\in{B}$, by definition of cartesian product $a\in{A} \wedge b\in{B} \Leftrightarrow (a,b)\in{A\times{B}}$, as desired.
The questions I arrived are the following:
- Did I use the logic correctly?
- If my proof is correct, how can I improve it?
- If they are logically equivalent, which one is more elegant to use in mathematics research?
As a notation in a paper, for example $$\forall a \in A: \forall b \in B: P(a,b)\tag{1}$$ is semantically the same as $$\forall (a,b) \in A \times B: P(a,b)\tag{2}$$
so in that sense is does not matter and they are eqeuivalent. But realise that $A \times B$ is a set theory notion, and that $(1)$ is the more "purely logic way" of formulating it. So I'd only use $(2)$ if $A \times B$ were also some object of interest in the context. I wouldn't introduce that set, if it were extraneous or just to express the formula with one quantor or some such reason.