There is a question that has been bothering me where the concept is confusing to me. Assume B is the set of all boys and G is the set of all girls. L(B,G) represents that B likes G.
$$\forall b \in B, \forall g \in G, \neg L(b,g) \rightarrow \exists! h \in G [L(b,h) \wedge \forall c \in B \neg(L(c,g) \wedge L(c,h))] $$
From this statement, wont it be that the predicate contradicts with the existential? I know that statement is true, because the prof said so. So the predicate is true and none of the boys like none of the girls. So how can their exist a h that is a girl that likes a boy? I'm confused on this part.
This is presumably meant to be interpreted with parentheses around everything after the first two quantifiers, like this: $$\forall b \in B, \forall g \in G, (\neg L(b,g) \rightarrow \exists! h \in G [L(b,h) \wedge \forall c \in B \neg(L(c,g) \wedge L(c,h))]).$$ So it's not asserting that if every boy dislikes every girl, then there exists a unique girl $h$ such that... It's asserting that for each particular boy $b$ and each particular girl $g$, if $b$ dislikes $g$ then there exists a unique girl $h$ such that...
Note that with your interpretation, it doesn't even make sense to ask whether the statement is true: the variables $b$ and $g$ would be free in the second half of the statement, so the truth of the statement would depend on what values you plugged in for them.