Given a function f from X to Y, if f is 1:1, then $\displaystyle f(A\cap B)=f(A)\cap f(B)$ for any subsets A, B of X. So in this situation, let f be the function from G to G where $\displaystyle f\,:\,g\mapsto x_j^{-1}gx_j$. So switching to exponential notation (you can change if you like):
$\displaystyle <e>=H\cap H^{x_ix_j^{-1}}$ implies
$\displaystyle <e>=<e>^{x_j}=(H\cap H^{x_ix_j^{-1}})^{x_j}=H^{x_j}\cap H^{x_ix_j^{-1}x_j}=H^{x_j}\cap H^{x_i}$