You can't possibly prove this because it isn't true. I don't know an easy finite counter-example off-hand, though I guess any you could do something like pick the cyclic Abelian group of order (30) which has normal subgroups with 5 3 and 2, elements which all have trivial intersection.
A nice infinite counter example comes from the wallpaper groups. P4MM contains C2MM P2MM and P4 as subgroups of index 2. They are all obviously normal subgroups but distinct from one another.