Smullyan's Drinking Principle/Semi-Formal Proof

Proof
Either everyone in the pub is drinking or someone in the pub is not drinking.

Suppose that everyone in the pub is drinking.

By True Statement is implied by Every Statement, the statement:
 * everyone in the pub is drinking

is implied by the statement:
 * $x$ is drinking

for any $x$ in the pub.

Since the pub is by assumption non-empty, there exists some $x$ in the pub.

Thus, there is some $x$ in the pub such that, if $x$ is drinking, then everyone in the pub is drinking.

Now suppose that there is some $x$ in the pub who is not drinking.

By False Statement implies Every Statement, if $x$ is drinking then everyone in the pub is drinking.

Thus, there is some $x$ in the pub such that, if $x$ is drinking, then everyone in the pub is drinking.