Well, logically, he does have /some/ equity if there is any in the house.
Granted, this is personal opinion, not legal, but if you wanted to show him how much/little of the equity he would be reasonably allowed to claim in the house, do this:
What you will need to know, to mathmatically figure out what he has put toward any equity that comes up would be how much of those 6 mortgage payments was on the principal, and how much was interest. The payments on the principal are what we're interested in. Let's say it was ~$200 a month. I have no idea, I'm just guessing here to show you numbers.
He put in the following:
$600 + ($200 * 6) == 1800
You put in:
$6000 + ($200 * 6) == 7200
That makes a total of $9000 'put into' the house.
1800 / 9000 = 20% (0.2)
7200 / 9000 = 80% (0.8)
so, say the house now has $10,000 in equity, you and he could show number wise he has a 'right' to $2,000.
If the principal on the mortgage was higher, then he would get more % in this, if it was lower, he would get less, but you have to remember there is principal payments that go towards equity as well.
As to what anyone is entitled to under the law, I have no idea, someone else might be able to get you better idea on that, but I think most would agree that this would be a rather fair way to figure out actual percentage of equity. Interest portion of the payments is a wash as it's the 'tax' on not paying for it all at once so to speak. If you really wanted to be complicated you may even be able to justify adjusting his part of the principal payment monthly to match his % of contribution, but that might be a little over the top.
Bottom line, just counting it from the down payment is likely to cause some issues, but if you show him you're taking into account payments he made monthly, then perhaps he may be a bit more reasonable with you.
Granted, you could argue that his half of the monthly mortgage was rent, but I can't say that everyone would see it that way.
Again, not a 'legal' opinion, but, I feel, a reasonable opinion from an outsider. Take it as what you will.