Q:

Can I replace conjunction with implication here?

Using a model where the domain is $\mathbb{N}$ and one binary predicate $p$ exists, that returns $true$ if its second argument is divisible by the first one, show that each two numbers have a biggest common divisor.
I came up with such an answer:
$\forall_{x}\forall_{y}\exists_{z}((p(z,x)\wedge p(z,y))\Rightarrow\forall_{v}((p(v,x)\wedge p(v,y))\Rightarrow p(v,z)))$
but in the textbook there is a conjunction instead of the first arrow. I can't tell if both answers are correct or not. I know that the two would have different meaning, but I can't figure out if there is something wrong with my approach.

A:

There is a law of propositional logic called Exportation :

$((P \land Q) \rightarrow R) \leftrightarrow (P \rightarrow (Q \rightarrow R))$.

But $((P \land Q) \land R)$ is not equivalent to : $((P \land Q) \rightarrow R)$.

