James,
In my opinion you would be taking a big risk unless you actually had the capital to close the initial deal successfully without the proceeds from your sale of your home. You should discuss the feasibility of funding with your loan officer first. The contingency clause is part and parcel of your original contract. It sounds like you are assuming you will close and have those assets to bring to your offer. Any offer you made me would require those assets be available first.
respectfully
Chuck Sparks

James,
Don't push your luck. Consider yourself fortunate to have sold your current home so quickly. Many, many people that are in your situation are finding it very difficult to even offer on a home until they have a contract on the home they have to sell. All that's happened here is that you are one step closer to satisfying the sales contingency. This does not entitle you to a lower price on the home you are buying. Be happy you've got a buyer for your home!

If you did it before you had an offer, and could and would close without selling the other house, the liklihood would be higher.

If you can't buy their house unless you close on your house, you could be risking your Earnest Money if you pull the contingency. It should be contingent on the closing of your current home, and you might want that protection all the way to the end.