The relation between "x and $x

There is a very simple relation between ""xj" and "$xj" (where "j" is some predicate containing "x").

(Think of it like this: when you push a "¬" through a quantifier in either direction, it flips into a quantifier of the other sort.)

So one can translate "Nothing is brown" either as "¬$x x is brown" or as ""x ¬x is brown".

Equally one can translate "Everything is brown" as either ""x x is brown" or as "¬$x ¬x is brown".

We can see also that the two suggested translations of "No cows are brown" are indeed equivalent. "¬$x[x is a cow Ù x is brown]"  is equivalent to ""x¬[x is a cow Ù x is brown]". But "¬[jÙy]" is equivalent to "[j®¬y]". (Check it by a truth-table.) So, ""x¬[x is a cow Ù x is brown]" is equivalent to ""x[x is a cow ® ¬x is brown]".