Intersection of Equivalences
Let $\mathcal R_1$ and $\mathcal R_2$ be equivalence relations on $S$.
Let $\mathcal R_3 = \mathcal R_1 \cap \mathcal R_2$.
Checking in turn each of the criteria for equivalence:
So, by Intersection of Reflexive Relations is Reflexive, so is $\mathcal R_3$.
So, by Intersection of Symmetric Relations is Symmetric, so is $\mathcal R_3$.
So, by Intersection of Transitive Relations is Transitive, so is $\mathcal R_3$.
Thus $\mathcal R_3$ is an equivalence.