Let $\mathbb{Z}$ be the set of integers $$\lbrace \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \rbrace.$$ Define a binary relation on $\mathbb{Z}$ by declaring that $a \equiv b$ if and only if $a - b = 2^i$ for some non-negative integer $i$.

Is $\equiv$ an equivalence relation? Prove that it is, or explain which parts of the definition of equivalence relation do not hold.