separable utility function and cross price effect
The following argument assumes that we are dealing only with interior solutions. Let $(x_1,x_2,x_3)$ be an optimal demand bundle at prices $(p_1,p_2,p_3)$ and income $m$ and assume that $(x_1,x_2,x_3)$ is an optimal demand bundle at prices $(p_1',p_2,p_3)$ and income $m$. We want to show that one of the following three cases holds: 


$x_2=x_2'$ and $x_3=x_3'$,
$x_2&lt;x_2'$ and $x_3&lt;x_3'$, or
$x_2&gt;x_2'$ and $x_3&gt;x_3'$.


What is not allowed to happen is that $x_2\leq x_2'$ and $x_3&gt;x_3'$ or $x_2&gt;x_2'$ and $x_3\leq x_3'$. Assume one of these cases happens, without loss of generality, we can assume it is the first.

By the strict concavity of $f_2$ and $f_3$, we have
$$MU_2(x_1,x_2,x_3)=\frac{\partial f_2(x_2)}{\partial x_2}\geq \frac{\partial f_2(x_2')}{\partial x_2}=MU_2(x_1',x_2',x_3')$$
and
$$MU_3(x_1,x_2,x_3)=\frac{\partial f_3(x_3)}{\partial x_3}&lt; \frac{\partial f_3(x_3')}{\partial x_3}=MU_3(x_1',x_2',x_3').$$ 
Therefore,
$$\frac{MU_2(x_1',x_2',x_3')}{p_2}\leq\frac{MU_2(x_1,x_2,x_3)}{p_2}=\frac{MU_3(x_1,x_2,x_3)}{p_3}&lt;\frac{MU_3(x_1',x_2',x_3')}{p_3},$$
in contradiction to the equimarginal principle.