Your solution claims that whenever there is someone who dislikes some vegetarian, the one doing the disliking is smart.
For example, suppose that indeed, everyone who dislikes all vegetarians is smart. Suppose further John is kinda stupid, and likes everybody except for Brian, who is a vegetarian. That is not a conflict, because John doesn't dislike all vegetarians: Henry is a vegetarian, and John likes him just fine. However your first-order formula is false in this situation.
It is false because the only thing that is needed to make a $\forall$ formula false is a single counterexample. And $x=\text{John}, y=\text{Brian}$ is a counterexample -- the body of the quantifier then becomes: $ \text{person(John)}\land\text{vegetarian(Brian)}\land\text{dislike(John,Brian)}\to\text{smart(John)}$ and here all the conjuncts to the left of the $\to$ are true, but the conclusion on the right is not.
I suspect you started out fine with something like $ \forall x\big[\text{person}(x)\land\text{dislikes-all-vegetarians}(x)\to\text{smart}(x)\big]$ but then you when wrong translating "dislikes-all-vegetarians". By itself it is $ \text{dislikes-all-vegetarians}(x) \equiv \forall y[\text{vegetarian}(y)\to \text{dislike}(x,y)]$ and we can insert it into the first formula to get $ \forall x\big[\text{person}(x)\land\forall y[\text{vegetarian}(y)\to \text{dislike}(x,y)]\to\text{smart}(x)\big]$ Note that here you cannot move the $\forall y$ out such that it is neighbor to the $\forall x$, because it is important for the meaning that $\forall y$ is subordinate to the $\rightarrow$ rather than the opposite.
We can insist on getting a logically equivalent formula where all quantifiers are out at the front (a "prenex formula"), but then the $\forall y$ becomes $\exists y$, which makes the result rather unrecognizable as a rendering of the original English sentence: $ \forall x \exists y \big[ \text{person}(x) \land [\text{vegetarian}(y) \to \text{dislike}(x,y)] \to \text{smart}(x)]\big]$ or $ \forall x \exists y \big[ \text{person}(x) \to \text{smart}(x) \lor [\text{vegetarian}(y) \land \neg\text{dislike}(x,y)]\big]$ (more or less "every person is smart unless (possibly) if there is a vegetarian he doesn't dislike").