Smullyan's Drinking Principle/Formal Proof

Proof
We have two choices:
 * $\forall y : \map D y$

and
 * $\neg \forall y : \map D y$

Suppose $\forall y : \map D y$.

By True Statement is implied by Every Statement:
 * $\map D x \implies \forall y : \map D y$

By Existential Generalisation:
 * $\exists x : \paren {\map D x \implies \forall y : \map D y}$

Now suppose:
 * $\neg \forall y : \map D y$

By De Morgan's Laws (Predicate Logic)/Denial of Universality:
 * $\exists y : \neg \map D y$

Switch the variable $y$ with $x$.

Thus, for some $x$:
 * $\neg \map D x$

By False Statement implies Every Statement, we have:
 * $\map D x \implies \forall y : \map D y$

By Existential Generalisation:
 * $\exists x : \paren {\map D x \implies \forall y : \map D y}$

Thus, $\exists x : \paren {\map D x \implies \forall y : \map D y}$ holds both when:
 * $\forall y : \map D y$

and when:
 * $\neg \forall y : \map D y$

concluding the proof.