$\exists!$ is $uniqueness\,quantifier$ which says for $exactly\,one$
a) $\exists !x P(x)\rightarrow \exists x P(x)$
The LHS says "For exaclty one $x ,\,P(x)$ is True", RHS says "For some $x,P(x)$ is True". If LHS is True, RHS is True, so this is always $True$
b) $\forall x P(x) \rightarrow \exists ! x P(x)$
The LHS says "For all $x, P(x)$ is True ", RHS says "for exactly one $x, P(x)$ is True".So, if LHS is True, RHS is not True. This can only be true when $LHS$ is $False$.
c) $\exists!x \neg P(x) \rightarrow \neg \forall x P(x)$
$\exists!x \neg P(x) \rightarrow \exists x \neg P(x)$
The LHS says "For exactly one $x, P(x)$ is not True", RHS says "For some $x,P(x)$ is not True". If LHS is True then RHS is also True. So, this is always $True$.
