Step 1: 

Y = A̅B̅C̅D̅ + A̅B̅C̅D + A̅BC̅D̅ + A̅BC̅D + AB̅CD + AB̅CD̅ + ABCD̅

Step 2: 

Y = A̅B̅C̅(D̅ + D) + A̅BC̅(D̅ + D) + AB̅CD + AB̅CD̅ + ABCD̅

Step 3: 

Y = A̅B̅C̅ + A̅BC̅ + AB̅C(D + D̅) + ABCD̅

Step 4: 

Y = A̅B̅C̅ + A̅BC̅ + AB̅C + ABCD̅

Step 5: 

Y = A̅C̅(B̅ + B) + AC(B̅ + BD̅)

Step 6: Simplify Further

Y = A̅C̅ + AC(B̅ + BD̅)

This is the final simplified expression:

Y = A̅C̅ + AC(B̅ + BD̅)