The theorem says: every planar graph has a vertex of degree at most 5.
In another question, the proof by contradiction for that is
Suppose that there exists a planar graph with all vertices having degree at least $6$. Then by the handshake lemma, $E = 3V$. If $G$ is planar, then $E \leq 3V - 6$, a contradiction.
In general, I'm familiar with the handshake lemma:
$2E = \sum deg(v)$
What I don't understand is, why can I substitute $\sum deg(v)$ with $6V$, so that the equation becomes $2E = 6V$ and simplifies to $E = 3V$. Why can I just replace the sum with $6V$ ?