For predicates, it is possible to pass multiple variables. They are called polyadic predicates in that case, for example, $P(x, y)$. But how about quantifiers. I have not seen an example of this but want to be sure.
Is there usage for something like:
$$∀(x∈A, y∈B)(x | y)$$
in the first-order logic?
I could provide arguments in the following manner:
If multiple domain sets are provided, then values could be passed to the quantifier in tuples taking one item from each domain to match each argument. Say $A = [1 2 3]$ and $B = [1 2 3 4]$. Then we send arguments in groups of: $(x=1, y=1)$, $(x=2, y=2)$, and $(x=3, y=3)$ and leave the last iteration out because there is no pair for y: $(x=undefined, y=4)$. Is this too far from convention?
Or should it just be:
$$∀x∈A(∀y∈B(x | y))$$
Or maybe even:
$$∀x∈A∀y∈B(x | y)$$
Apparently b) and c) would give same truth value, but a) differs dramatically from the last two...
$|$ is just an arbitrary operation between $x$ and $y$.