Translating predicate logic statements

Question

Consider the following predicates over the domains $B$ of all bunny rabbits, $M$ of all magicians, and $J$ of all jazz musicians.

$F(x; y)$: person $x$ sent a friend request to person $y$.

$D(x; y): x$ made $y$ disappear.

$H(x; y): x$ enjoys being made to disappear by $y$.

a) $\forall a\in B\ \neg(\exists c\in B\ a\ne c\land D(a,c))\iff\forall b\in M\ H(a,b)$

I have: every bunny rabbit cannot make some other bunny rabbits disappear if and only if every bunny rabbit enjoys being made to disappear by magicians.

Not sure if this is right because I'm having a little trouble translating $\lnot (\exists c \in B, a \neq c \land D(a,c))$ into English.

Answer 1

The best way to approach these problems is to literally say them out: $$\forall a\in B\ \neg(\exists c\in B\ a\ne c\land D(a,c))\iff\forall b\in M\ H(a,b)$$

For all bunnies $a$ it is not true that (there exists a different bunny $c$ it made to disappear) iff (for all magicians $b$ it likes being made to disappear by it).

Then simplify:

A bunny that has not made other bunnies disappear likes being made to disappear by all magicians and vice versa.