On the $\pmb{\forall}\pmb{\exists}$-theories of free projective planes

Studying the elementary properties of free projective planes of finite rank, we prove that for $m>n$, an arbitrary $\forall\exists\forall$-formula $\Phi(\bar{y})$, and a tuple $\bar{u}$ of elements of the free projective plane $\frak{F}_n$ if $\Phi(\bar{u})$ holds on the plane $\frak{F}_m$ then $\Phi(\bar{u})$ holds on the plane $\frak{F}_n$ too. This implies the coincidence of the $\forall\exists$-theories of free projective planes of different finite ranks.