Let $N\subset M$ be a substructure. Show that for every quantifierfree L-Formula(That is, a formula without quantifiers) $\phi$ with variables $x_1, \dots ,x_n$ and for very $n$-tuple $m_1, \dots ,m_n$ of elements of $N$, we have: $$N \models \phi(m_1,\dots ,m_n) \text{ if and only if } M\models\phi(m_1,\dots ,m_n) $$
As often is the case, the answer intuitively clear, but hard to formalize. My attempt so far has been: Without quantifiers, the statements must consist of either and, or, not, disjunctions, the absurdity, or an equality. All these symbols hold for the elements of $M$ themselves and are not dependent on other elements, that is $x_1 \land x_2 = \bot$ is true regardless of the models that surround $x_1$ and $x_2$. For example, the fact that $(2 * 2 = 4)$ is true, depends only on $2$ and $4$, not on wether this in in $\mathbb{N}$ or $\mathbb{R}$ Due to the fact that all other symbols only depend on the elements and not the set in which they are contained, it must so that if such properties hold in $N$, then it must hold in $M$ as well.
With quantifiers it is easy to find a counter example, take $M=\mathbb{R}$ and $N = \mathbb{N}$. Then let $f_m(k)$ be multiplication and let $\phi=\exists m(f_m(m) = 2)$ which is obviously false in $N$, but true in $M$.
However, this answer is very intuitive. While intuitively clear, I don't really know how to define "Only depends on the elements themselves" properly. How can I make this argument more rigorous?