The OP’s plan is totally flawed for several reasons:
The “best” tires (i.e.–the ones with the most tread depth) are supposed to be mounted on the REAR wheels of a FWD car. Otherwise, you risk loss of rear traction on wet roads. If you don’t believe me, check the websites of Michelin, Bridgestone, and other major tire companies for verification. If your tire shop mounted the new tires on the front, then they are not very knowledgeable about modern tire protocols.
Modern technology winter tires (the term “snow tires” now represents old technology) are designed to be mounted in sets of 4. Having vastly different amounts of traction on front and rear wheels during winter conditions is the perfect storm (pun intended) for spin-outs and skids, and your plan of mounting “snow tires” on just the front would give you vastly different amounts of traction from front to rear. This is a particularly bad idea.
Winter tires have a tread compound that is designed to remain soft in frigid temperatures–unlike all-season tires and the old-technology snow tires. That is part of the technology of modern winter tires that allows them to provide far better traction than the old-technology “snow tires”. However, the rubber compound of a winter tire will wear VERY rapidly in spring and summer temperatures. Winter tires should be de-mounted as soon as the threat of snow and ice is over for the season.
So–as texases suggested, either bite the bullet and buy new matching all-season tires for the rear now, or wait until the beginning of winter and buy a set of 4 winter tires. We all want to save money, but your plan could wind up costing you much more in the long run–if and when it results in a collision.