Modular Arithmetic and the Distributive Property

Question

I know that we can't apply the distributive property when doing modular arithmetic, e.g $$(a+b) \text{ mod } c \neq a\text{ mod c} + b \text{ mod c}$$

But does it hold that

$$\bigg((a+b) \text{ mod }c\bigg)\text{ mod c} = \bigg(a\text{ mod c} + b \text{ mod c}\bigg)\text{ mod c}$$