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Simultaneous propositional substitution givesLet $K=\neg (X\wedge Y)$. The first disjunct equals $K\vee \neg (X\vee Y)$. By De Morgan's laws, $\neg (X\vee Y)=\neg X\wedge \neg Y$ implies $K=\neg X\vee \neg Y$, so absorption reduces that first disjunct to $K$. The second disjunct is $\neg Y\wedge K$ and is also absorbed by $K$. ThereforeThe resulting Boolean function is false only when both $X$ and $Y$ are true, providing a four-row truth table check.

Apply the propositional substitution simultaneously to the atomic variables:By distributivity and the complementary literals $Y,\neg Y$,The other disjunct simplifies using logical implication and absorption:Thus $\phi S\equiv X\vee \neg X\equiv \top$, a tautology.

A propositional substitution maps each atomic propositional variable to a propositional formula and extends recursively through every logical connective. Replacements are simultaneous: a variable appearing inside a replacement formula is not replaced again in the same substitution step. Since propositional logic has no variable-binding quantifiers, this operation has no variable-capture problem.