Erosion compotition rule $\left ( f \ominus g \right ) \ominus h = f \ominus \left (g \oplus h \right ) $
Morphological Erosion is not associative as dilation. Much like addition is associative but not subtraction. Consider the following basic definitions first:$$(A \ominus B)^C = A^C \oplus \widetilde{B} \tag{1}$$
$$(A \oplus B)^C = A^C \ominus \widetilde{B} \tag{2}$$

where $\widetilde{B}$ is the reflection of kernel $B$ and $A^C$ is the compliment of image $A$, inverse of image $A$. Now let us evaluate the expression in the question:
$$(A \ominus B) \ominus C = ((A\ominus B)^C \oplus \widetilde{C})^C \tag{using eq. 1}$$

$$(A \ominus B) \ominus C = ((A^C\oplus \widetilde{B}) \oplus \widetilde{C})^C \tag{using eq. 1} $$

$$(A \ominus B) \ominus C = (A^C \oplus (\widetilde{B} \oplus \widetilde{C}))^C \tag{since dilation is associative}$$

$$(A \ominus B) \ominus C = A \ominus \widetilde{(\widetilde{B} \oplus \widetilde{C})} \tag{using eq. 2}$$

$$(A \ominus B) \ominus C = A \ominus (B \oplus C) \tag{distributing the reflection}$$

Hope that answers the question