In my lecture notes it's written that the negation of
For all lines $l$ and for all points $P$ not on $l$, there exists a unique line $m$ passing though $P$ and parallel to $l$
is
For all lines $l$ and for all points $P$ not on $l$, there exists no line $m$ passing though $P$ and parallel to $l$
or
For all lines $l$ and for all points $P$ not on $l$, there exists more than one line $m$ passing though $P$ and parallel to $l$
but I really think it's wrong to deny it this way, instead of ''For all lines $l$ and for all points $P$'', it should be ''There exist at least a line $l$ and a point $P$...''.
Am I wrong ?