(Perhaps not a direct answer to your question, but this might be helpful.)
In my opinion this is most easily understood as a specific instance of
$$(0) \;\;\; x = y \;\Rightarrow\; (P(x) \Rightarrow P(y))$$
where $\;P(\cdot)\;$ is any boolean expression of one variable. And in turn this is a weaker form of
$$(1) \;\;\; x = y \;\Rightarrow\; (P(x) \equiv P(y))$$
which Dijkstra calls Leibniz' rule, and which the Wikipedia page on the topic calls "The indiscernibility of identicals".
Now $(1)$ in plain English: "If $\;x\;$ equals $\;y\;$, then whatever is true of $\;x\;$ is true of $\;y\;$, and vice versa." (If you leave out the "and vice versa" part, of course you get $(0)$.)