Logic: some basic plane geometry

Logic: some basic plane geometry

Suppose you've got the language of some basic plane geometry, i.e. two
1-place relation symbols $P$ and $L$ for point and line and one 2-place
relation symbol $I$ for point $x$ lies on line $y$. Now, how is it
possible to express the following axiom in this language: for every line
$l$ and point $x$ (which is not on $l$), there is a unique line $m$
through point $x$.
This is what i've got: $\forall l,x$ $\neg I(x,l)\rightarrow \exists m$
$I(x,m)$.
Is this a correct sentence to begin with, and how could I express the
uniqueness of the line $m$?