Translating "nobody likes a sore loser"

Question

I tried translating the utterance "nobody likes a sore loser" into FOL.

Using the following glossary,

• $S(x)$: $x$ is a sore loser
• $L(x, y)$: $x$ likes $y$

I came up with the following two translations:

1. $\forall x[ \neg\exists y( S(y) \land L(x, y) )]$.
2. $\forall y[S(y) \implies \neg\exists x L(x, y)]$.

But only (2) matches the translation given in the textbook. I wasn't sure if that meant that (1) was incorrect or if (1) and (2) were equivalent, so I checked for equivalence:

1. $$\forall x[ \neg\exists y( S(y) \land L(x, y) ) ] \iff \forall x \forall y[ \neg( S(y) \land L(x, y) ) ]$$.
2. $$\begin{align*} \forall y[ S(y) \implies \neg\exists x L(x, y) ] &\iff \forall y( \neg\exists x [ S(y) \implies L(x, y) ] ) \\ &\iff \forall y \forall x [ \neg (S(y) \implies L(x, y)) ] \\ &\iff \forall y \forall x [ S(y) \land \neg L(x, y) ] \\ &\iff \forall x \forall y [ S(y) \land \neg L(x, y) ] \end{align*}$$.

But $\forall x \forall y [ S(y) \land \neg L(x, y) ]$ is not equivalent to $\forall x \forall y[ \neg( S(y) \land L(x, y) ) ]$. So, it turns out that (1) and (2) are not equivalent, which means they must represent different claims. But if (2) represents the claim "nobody likes a sore loser," then what claim does (1) represent?

Answer 1

Like @peterwhy mentioned in comment, you cannot just pull $\lnot\exists x$ out of the bracket, only have $$S\to\lnot\exists xL(x)\equiv S\to\forall x\lnot L(x)\equiv\forall x(S\to \lnot L(x))$$ However, both translation are correct! Let's show they are equivalent:

\begin{align*} &\forall x(\lnot\exists y(S(y)\land L(x,y)))\\ \equiv&\forall x\forall y\lnot(S(y)\land L(x,y))\\ \equiv&\forall x\forall y(\lnot S(y)\lor \lnot L(x,y))\\ \equiv&\forall y(\lnot S(y)\lor \forall x\lnot L(x,y))\\ \equiv&\forall y(\lnot S(y)\lor \lnot \exists x L(x,y))\\ \equiv&\forall y(S(y)\to \lnot \exists x L(x,y))\\ \end{align*}