Flow in a network: Conservation of flow definition

Question

This might be too easy... But I just don't get it.

I've been reading about flow in networks and I stumbled upon this definition in wikipedia: https://en.wikipedia.org/wiki/Flow_network

$\sum\limits_{w\in V} f(u,w) = 0 $ $(\forall u \in V-\{s, t\})$

That implies $\sum\limits_{(u,v)\in E} f(u,v) = \sum\limits_{(v, z)\in E} f(v,z)$

It sounds trivial, but how does that implication work? The flow is 0 when there is no edge. So I think I can rewrite the first sum to:

$\sum\limits_{w\in V} f(u,w) = 0 \iff \sum\limits_{(u,v)\in E} f(u,v) = 0 $ for a node $u$

That would result in every flow being zero, wouldn't it? What am I doing wrong?

Thanks in advance :)