Given any sentence in sentential logic with two variables ($\mathbf{P}$ and $\mathbf{Q}$), is it possible to reduce it to an equivalent sentence where each variable is only invoked once?
As an example off the top of my head, let's take a sentence
$\big((\mathbf{P}\lor\mathbf{Q})\land\mathbf{Q}\big)\land\big((\mathbf{P}\land\mathbf{Q})\lor\lnot\mathbf{P}\big)\;.$
I've just derived that it's equivalent to $(\mathbf{Q}\lor\lnot\mathbf{P})$, which is a sentence that only invokes $\mathbf{P}$ and $\mathbf{Q}$ once each. Is this applicable to any complex sentence with two variables?
My thought process is that, in a truth table with two variables, the only possible values of a sentence are all true, three true and one false, two of each, one true and three false, and all false. These line up with $\mathbf{T}$, $\mathbf{P}\land\mathbf{Q}$, $\mathbf{P}\leftrightarrow\mathbf{Q}$, $\mathbf{P}\lor\mathbf{Q}$, and $\mathbf{F}$, respectively. The order of the $\mathbf{T}$'s and $\mathbf{F}$'s in the table may differ, but that can be accounted for by applying negation to one or both of the operands.