First, I’m going to talk about Leibniz’ Law of Identity (LI), and then I’m going to make an application to Christianity. So hold on with me if you can! Many people find LI intuitive, and here it is:
For any objects x and y, if x and y are identical, then for any property P, x has P if and only if y has P.
This is to say that LI means that if something is identical to something else, they share all the same properties. But an immediate problem presents itself. Let’s take me for an example, and something trivial about me. I currently have the property typing on a keyboard. However, I can lack this property (in fact, not two hours ago I did lack this property). Or take the number of hairs on my head, or how tall I am, etc. All of these seem to be contingent, and in some cases, the property fell away completely or was acquired later. But if LI is correct, this means that the person I was two hours ago is not the same person typing this—and this in virtue of a great many things, not the least of which are typing and being fifteen feet away from Jodi and many others. But why should my mere spatial location in relation to Jodi dictate my existence (or the lack of my predecessor’s existence)? Something has gone wrong.
So perhaps we interpret LI to be inclusive of worlds and times. So in terms of personhood, x is the object that is my person and only includes essential properties, and y is my world-indexed person. But we still have a problem. Because while y-me contains all of the same properties that x-me does, the reverse is not true. And this means, according to LI, we’re not the same. So what we need is likely to include possible worlds and tensed language into both sides of the deal, and now we have a solution.
LI’ For any objects x and y, if x and y are identical, then for any property P, any world W, and any time t, x has P in W at t if and only if y has P in W at t.
And now this may help. How can we apply this in the Christian world? The identity of Jesus may be an area this works in. How is it that Jesus can be God? Isn’t this a contradiction? Let’s view Jesus as God the Son. Now ask yourself if God the Son and Jesus can satisfy LI’. It seems that he can. God the Son had a body that died on the cross; Jesus had a body that died on the cross, and so on. It’s an interesting tidbit on the law of identity and who Jesus is!
 This is all inspired by and taken from Thomas McCall’s note in Philosophia Christi.