Let$\ H$ be a hyperinteger. If$\ f(n)=g(n)$ is true for all$\ n \in \mathbb{N}$, will it be so for all$\ H$?

Question

I do know that all true first order statements in$\ \mathbb{N}$ are also valid in$\ \mathbb{N^*}$, so for example$\ \sin(H \pi) =0$. As a consequence, my question is equivalent to: is$\ f(n)=g(n)$ a first order statement? (This probably has a very easy answer, but I know nothing about first order logic, and reading through the Wikipedia page hasn't really helped me)