Bounded Universal Quantification of Provable Arithmetic Formula is Provable

Theorem
Let $\map \phi x$ be a WFF in the language of arithmetic with $1$ free variable.

Let $\tau$ be a term in the language of arithmetic with no variables.

Let $\sqbrk a$ denote the unary representation of $a$.

Suppose that, for every $n \in \N$ such that $\N \models \map \phi {x \gets \sqbrk n}$:
 * $\map \phi {x \gets \sqbrk n}$

is a theorem of minimal arithmetic.

Suppose also that:
 * $\N \models \forall x < \tau: \map \phi x$

Then:
 * $\forall x < \tau: \map \phi x$

is a theorem of minimal arithmetic.

Proof
Let $t = \map {\operatorname{val}_\N} \tau$ be the value of $\tau$ under the natural numbers.

By definition of value of formula:
 * $\N \models \map \phi {x \gets x_0}$

for every $x_0 < t$.

Therefore, :
 * $\map \phi {x \gets \sqbrk {x_0} }$

is a theorem, for every $x_0 < t$.

Additionally, by Lower Section of Natural Number is Provable:
 * $\forall x: x = 0 \lor x = \map s 0 \lor \dotso \lor x = \sqbrk {t - 1} \lor \neg {x < \sqbrk t}$

is a theorem.

The result follows from False Statement implies Every Statement and Proof by Cases.