Proposition: $\textit{Let M and N be sets. Prove that $M \cap N = M$ iff $M \subseteq N$}\\$

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My Proof:
Going from the definition of set intersection and subsets, the above statement can be rewritten as follows:
$$\{a: (a \in M \wedge a \in N) \equiv a \in M \} \equiv \{a: a \in M \Rightarrow a\in N\}.$$
Now let $p = a \in M$ and $q = a \in N$. This allows us to further rewrite the above statement in predicate logic:
$$((p \wedge q) \equiv p) \equiv (p \Rightarrow q).$$
This is a tautology, as shown by the truth table below.

[![enter image description here](https://i.stack.imgur.com/eKT4w.png)](https://i.stack.imgur.com/eKT4w.png)

Therefore the intersection of the sets M and N equal N if and only if M $\subseteq$ N.

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Is my proof even correct? My math professor doesn't find it sound to convert this statement into predicate logic and go from there, he just breaks apart the if and only if into two implications, and then goes by the definition of subsets. Which approach is better? Am I not employing the correct definition of set equivalence in terms of logic?