I think what was meant was that . If that is what was meant, it is always true- those are different notations for the same thing.
If those really are then I would guess that they refer to "small" changes in x, y, and f and so that is an approximation to but is exact in the case that f is a linear function of x and y.
A third possibility is that your teacher was saying that which is the same as saying that - that is that the "mixed" second derivative are equal- independent of the order of differentiation. That is true as long as the second partial derivatives are continuous.