Exactly what you are proposing has been done by thousands of homeowners, in my opinion, who have essentially the exact setup you have.
Will it work.....more or less.
Is it legal...definitley not. You do not have a vent. That waste stack cannot serve as the vent. One way to make it legit is to vent with an air admittance valve ( Studor valve) which may or may not be allowed under your code.
Other plumbers on here will disagree...but my advice: just DO IT! Be prepared to remove it all, if you decide to put the house on the market.