Recall this common definition:

~ is an equivalence relation on a set S if:
1. a~a for all a in S (reflexive)
2. a~b implies b~a for all a and be in S (symmetric)
3. a~b and b~c implies a~c (transitive)

Theorem: A relation that is symmetric and transitive is reflexive (and therefore an equivalence relation)
Proof:
By symmetry, we have a~b implies b~a.
Combining with transitivity, we have a~b and b~a implies a~a, which is the definition for reflexivity. QED