It doesn't work that way.
The new mortgage would only cover a percentage of the appraised value of the new property.
What you're talking about, perhaps, is called a blanket mortgage--a mortgage that covers two or more properties.
However, I'm not sure you'd see any savings from your plan. Your way, you'd have one huge mortgage on the new property and a paid-off second property. In the conventional way, you'd have two mortgages adding up to the same amount.
Recognize that you may not have to put 20% down on the new home. With FHA, for instance, you'd only have to put 3.5% down. And there probably are programs that would allow you to put, say, 10% or 15% down. That, in turn would reduce your required downpayment, while giving you more money to pay off the mortgage on your current home.
Check with a Realtor or lender for more information (and possibly other strategies to help you accomplish your goal).
Hope that helps.