You have to requalify.
As already stated, the proceeds from the sale of your current property are used to pay off your current loan. That loan--that mortgage--is dead. Kaput. Period.
However, if you do end up with some cash, you can roll that over into the new property, reducing the needed amount of the new mortgage.
It kind of sounds as if you're trying to "flip" properties. If so, though, you're doing it all wrong. You're not leaving yourself nearly enough "margin." And the transaction costs will eat you alive. Further, that's assuming that you really were able to buy a house for $45,000-$75,000 under market.
Also, I'm confused. You say the house is listed at $220,000 but you're planning to sell if the price hits $250,000. What happens if, today, someone makes you an offer of $220,000? Would you sell? Technically, you wouldn't have to (though you might still owe your Realtor a commission), but it doesn't seem ethical to list a home at $220,000 if you're really not going to sell unless you can get $250,000 for it.
One other thing--and check with a loan officer or mortgage broker on this: Lenders don't like to make short-term loans. If you were lucky enough to get an offer tomorrow, a lender might be reluctant to give you a loan on the new house you'd like to buy.
And--if you bought a home for $175,000, why is your next goal a home for $180,000-$190,000. It probably would have been easier just to have gone with the larger purchase initially. And if you didn't quite qualify, maybe you could have saved the $5,000-$10,000 you'd have needed. Consider: The cost of selling your home for $250,000 (with commissions, closing costs, etc.) probably will run $15,000-$20,000. You'd have been better off just spending the $5,000-$10,000 in the first place.
Bottom line: Obviously, you've got some strategy or goal. And that's fine. But I think you might be able to benefit from a bit of advice on how to best get there.
Hope that helps.