Ok to keep two snow tires on front wheels and two all weather tires on back?

What is your question?

I recently went to get my oil changed and found out there was splitting on the inner side of the two back snow tires. I looked at it and since the tires were about 3 years old, I decided to replace them. The head mechanic told me he was going to order me all weather tires and put them on the next day. He did so and the car drives great. However, I am concerned that once winter hits, the car is going to start spinning out easily since the two back tires aren't snow tires (my dad has had issues with this when he still had this car). The mechanic told me it shouldn't be an issue however I would like a second opinion from anyone who can help! Especially since I could bring the tires back and then upgrade to two more snow tires.