Does the axiom imply its dual? Suppose we have the following axiom for the projective plane:

Axiom: If a projectivity leaves invariant each of three distinct points on a line, it leaves invariant every point of the line.

The dual of this axiom is the following statement:

Dual: If a projectivity leaves invariant each of three distinct lines, which are concurrent at a point, it leaves invariant every line passing through that point.

Now I need to prove that Axiom $\Rightarrow$ Dual.
My wrong attempt at the proof: Let the three lines $a,b,c$, concurent at $O$ be the lines left invariant by the projectivity, and $l$ be any other line through $O$. If we can prove that three points on $l$ are left invariant by this projectivity, then we will have that all the points on $l$ are left invariant by the projectivity, which would imply that $l$ itself is left invariant, and we will be done, because we have taken any arbitrary line through $O$.
Now we know that $O$ is left invariant by the projectivity. For any other point, say $P$, on $l$, let a line passing through $P$ other than $l$, intersect $a, b$ and $c$ at the points $A,B$ and $C$, respectively. Now if $A,B,C$ are left invariant by the projectivity, then $P$, a point on the line $AB$, will also be left invariant. Since $P$ is any arbitrary point on $l$, this is true for all points on $l$ and we are done.
However the lines $a,b,c$ being invariant does not imply that the points $A,B,C$ will be left invariant by the projectivity. That is where I am stuck.

Please share any insight to nudge me in the right direction.

You can think of a projectivity as an alternating sequence of points and lines and the sequences of perspectivities they represent. For a simple example, $(\ell_1,P_2,\ell_3)$ represents the perspectivity between lines $\ell_1$ and $\ell_3$ with center $P_2$.  And ($P_2,\ell_3,P_4$) is the perspectivity with axis $\ell_3$ between pencils at $P_2$ and $P_4$. ($\ell_1,P_2,\ell_3,P_4,\ell_5$) is a projectivity $\ell_1\rightarrow\ell_5$ that is the composition of two perspectivities $\ell_1\rightarrow\ell_3\rightarrow\ell_5.$
Let $(P_1,\ell_2,\dots,P_1)$ be a projectivity that leaves three lines in the pencil at $P_1$ invariant, but not the entire pencil.  Then consider $(\ell_2,\dots,P_1,\ell_2)$.  This is possibly already more than the nudge you've requested, but I'll leave the final step as a spoiler.

 $(\ell_2,\dots,P_1,\ell_2)$ is a projectivity that violates the Axiom.

