x could be 1.5 and y could be 1.5 and
[1.5 + 1.5] =  = 3 is not equal to [1.5] + [1.5] = 3
I don't know how you work in logic so I can't write a formal proof for you.
Recall that the floor of a real number x, i.e. [x], is defined to be the largest integer d such that d <= x
Proposition. For any real numbers x and y, [x + y] >= [x] + [y]
(a) Explain briefly why it's not true if you replace >= with =
(b) Prove that this claim is true.
Notice that the definition of floor folds together two properties. If t is the floor of z, then (i) t <= z and (ii) for any integer d, if d <= z, then t >= d.
It may help to give short names (e.g. like d) to expressions like [x + y].