In a recent question I was investigating (I will not disclose the question here, but will provide some analogous examples) I was trying to apply a certain restricted concept to a broader context. This property had the form
\begin{align} \forall_{x,\in X}[\alpha(x)] \end{align}
But the variable $x$ appears twice in the formula for $\alpha$, and thus we can make a new statement $\alpha'$ which depends on two variables and satisfies:
\begin{align} \forall_{x}[\alpha'(x,x) \iff \alpha(x)] \end{align}
So it is essentially the same statement as $\alpha$ but viewing the two appearances of $x$ in $\alpha$ as separate. This then allows us to create a new and weaker condition of a set, namely
\begin{align} \forall_{x\in X}\exists_{y\in X}[\alpha'(x,y)] \end{align}
Which is clearly weaker than the first condition, since if $\alpha(x)$ is true for some specific $x$ then $\alpha'(x,x)$ is true. The process can also sometimes be done in reverse to obtain a significant strengthening of a statement, for example in the definition of continuity of a function on $\mathbb{R}$, we have something of the form
\begin{align} \forall_{\varepsilon>0}\exists_{\delta>0}[\alpha(\varepsilon,\delta)] \end{align}
If instead we quantified
\begin{align} \forall_{\varepsilon}[\alpha(\varepsilon,\varepsilon)] \end{align}
Then the class of functions satisfying this will still be continuous, but they now have some other restrictions (Probably something resembling a restriction on their derivative, but a bit more nuanced)
*Note that this strengthening was also accompanied by a weakening of the domain from which $\varepsilon$ is taken, we don't need to worry about $\delta$ being negative when $\varepsilon$ is positive anymore.
My question is, is this a technique that has shown up enough in the past that it has a name? And what are some other interesting instances that can be seen as a version of this?