Prove that if a formula $\phi (v_1, v_2,...v_n)$ is satisfied in the substructure $\mathcal M$, then it is satisfied in structure $\mathcal N$

Question

Assume $\mathcal M \subseteq N$ structures for signature $S$. $\mathcal M$ is a substructure of $\mathcal N$. Let $\phi(v_1, \cdots v_n)$ be a formula without quantifiers. Prove by induction on building of $\phi$ that for every $b_1 \cdots b_n \in M$, $\mathcal M \vDash \phi(b_1,\cdots b_n) \iff \mathcal N \vDash \phi(b_1, \cdots, b_n)$

So I began proving it by induction. I proved that this occurs if $\phi$ is an atomic formula of the type $t_1=t_2$ or a relation $R$. Now I'm not so sure how to prove this for $\phi: \phi_1 \land \phi_2$. Any hints will be great!

Answer 1

Expand the definition of $\mathcal{M}\models\phi_1\land\phi_2$ and use the induction hypothesis:

\begin{align}\mathcal{M}\models\phi_1\land\phi_2&\Leftrightarrow\mathcal{M}\models\phi_1 \text{ and } \mathcal{M}\models \phi_2\\&\Leftrightarrow\mathcal{N}\models\phi_1 \text{ and } \mathcal{N}\models \phi_2 \qquad \text{ (by induction hypothesis)}\\&\Leftrightarrow \mathcal{N}\models\phi_1\land\phi_2\end{align}

And the same way for $\phi=\neg\phi_1$