How do you defeat golbez in final fantasy iv DS?

First I found a save point, saved, then went to fight Golbez. First in the battle, you'll attack him, but it won't do much damage.(No matter how high your level is, trust me)Then Golbez will summon a "Shadow Dragon".(or something along the lines of that,But It is some kind of Shadow dragon)Golbez will use the shadow to kill all of your people, but he saves Cecil for last. But when he's about to kill Cecil,Rydia comes in with her "Mist Dragon" to kill the shadow dragon.Golbez gets pretty peeved(mad)and changes what he's weak to. I used a besitary item to see what he's weak to then used Rydia's summon spirit that is the same as the weakness.(ex.If Golbez is weak to fire, use Ifrit, the fire spirit.)After you attack him once doing this strategy, he'll change his weakness again. Just do the strategy above 3-4 times and BAM you defeat Golbez for now.