How to prove that $f(x) = x ? y : z$ is basis of $\mathcal{P}$?

Question

$\mathcal{P}$ is the class of preserving 0 and 1. I expressed $x \lor y$ like $f(x, x, y)$ and $x \land y$ like $f(x, y, x)$. I also expressed $maj_3(x, y, z) = xz \oplus xy \oplus zy$ and $x \oplus y \oplus z \oplus xy \oplus xz \oplus yz \oplus xyz$. To finish the proof I need $\oplus_3(x, y, z) = x \oplus y \oplus z$, so I would be able to express all Zhegalkins polynomial with even quantity of terms and without 1, which makes exactly $\mathcal{P}$, but I can't find a way to express $\oplus_3(x, y, z)$.