Smart replacement everywhere?

eq = f == x^3 + 6/x^3; 
Reduce[{eq, x^3 == z}]

(x == z^(1/3) || x == (-(-1)^(1/3))*z^(1/3) || 
  x == (-1)^(2/3)*z^(1/3)) && 
   z != 0 && f == (6 + z^2)/z