Let our language be $\{+,\cdot,<,0,1\}$. Let $T_n$ be the set of of true statements in that language (true in the standard model $\mathbb{N}$) that have at most $n$ bound variables and no other variables. Let $P_n$ be the set of statements in that language that are provable in first-order Peano arithmetic that have at most $n$ bound variables and no other variables.
Let me give some examples. $0<1$ is in both $P_0$ and $T_0$. $(\forall x) (0+x)=x$ is in $P_1$ and $T_1$. Also, duplicate variables are counted more than once, like for example $((\forall x) x=x \vee (\forall x) (x+0)=x)$ would be in $T_2$, not $T_1$.
Certainly, $P_n$ is a subset of $T_n$. Also, $P_0 = T_0$. My question is, what is the least $n$ such that $P_n$ is a proper subset of $T_n$?