quantum field theory - Why is the functional integral of a functional derivative zero?

I'm reading Quantum Field Theory and Critical Phenomena, 4th ed., by Zinn-Justin and on page 154 I came across the statement that the functional integral of a functional derivative is zero, i.e.

$$\int [d\phi ]\frac{\delta F[\phi]}{\delta\phi^{\alpha}(x)} = 0$$ for any functional $F[\phi ]$.

I would be most thankful if you could provide a mathematical proof for this identity.