Show that $∃x[∀yLxy ⇔ ∃z¬Lxz]$ is a contradiction. I know that to prove contradiction, I need to prove that $∃x[∀yLxy ⇔ ∃z¬Lxz]$ can derive $y$ & not $y.$
I got stuck at formulating the contradiction, because I don't know if $∀yLxy$ is the same as $∀zLxz$ intuitively. I think these two are different, but the contradiction I'm thinking about requires these two to be the same.
 A: The statement $$\exists x[\forall y Lxy \leftrightarrow \exists z \neg Lxz]$$ says that there is some object $x$ for which $$\forall y Lxy \leftrightarrow \exists z \neg Lxz$$ is true. So, if we call that object $a$, we get that
$$\forall y Lay \leftrightarrow \exists z \neg Laz$$
and by the quantifier negation law that means:
$$\forall y Lay \leftrightarrow \neg \forall z Laz$$
and now you see the contradiction: the left side says that $a$ stands in relation $L$ to every object, and yet the right side denies exactly that
(and yes, the fact that $y$ is used on the left side, and $z$ on the right side makes no difference: just as $\forall x P(x)$ and $\forall y P(y)$ say exactly the same thing .. variables are just 'dummy placeholders')
Now, not knowing how exactly (formal proof? truth tree? formal semantics? A more informal argument of the kind I just showed?) you are being asked to show that the statement is a contradiction, I can't help you any further, but hopefully you'll be able to take this line of reasoning and transform it into the kind of proof you are supposed to create.
