ILL 1967 Problem 50

The function $\varphi(x,y,z)$ defined for all triples $(x,y,z)$ of real numbers, is such that there are two functions $f$ and $g$ defined for all pairs of real numbers, such that $$\varphi(x,y,z) = f(x+y,z) = g(x,y+z)$$

for all real numbers $x,y$ and $z.$ Show that there is a function $h$ of one real variable, such that $$\varphi(x,y,z) = h(x+y+z)$$

for all real numbers $x,y$ and $z.$