One Threshold for Finitely Many Monotone Conditions and for Tails

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $I$ be a finite set, and let $\mathbb{N}$ and its order $\le$ be as in Arithmetic and Order of the Natural Numbers. Both clauses below are statements of the metatheory: they concern conditions expressed by formulas.

¶ Let $P(p,N)$ be a condition on the set variables $p$ and $N$, possibly with parameters, that is expressed by a predicative formula, and that is monotone: for all $p\in I$ and $N,N'\in\mathbb{N}$ with $N\le N'$, $P(p,N)$ implies $P(p,N')$. If for every $p\in I$ there is $N\in\mathbb{N}$ with $P(p,N)$, then there is one $N\in\mathbb{N}$ with $P(p,N)$ for every $p\in I$.

¶ Let $Q(p,n)$ be a condition on the set variables $p$ and $n$, possibly with parameters, that is expressed by a predicative formula. If for every $p\in I$ there is $N\in\mathbb{N}$ such that $Q(p,n)$ holds for every $n\in\mathbb{N}$ with $N\le n$, then there is one $N\in\mathbb{N}$ such that $Q(p,n)$ holds for every $p\in I$ and every $n\in\mathbb{N}$ with $N\le n$.