Clarification regarding Drinker's paradox This is the informal proof of Drinker's paradox

The proof begins by recognising it is true that either everyone in the
  pub is drinking (in this particular round of drinks), or at least one
  person in the pub isn't drinking.
On the one hand, suppose everyone is drinking. For any particular
  person, it can't be wrong to say that if that particular person is
  drinking, then everyone in the pub is drinking — because everyone is
  drinking.
Suppose, on the other hand, at least one person isn't drinking. For
  that particular person, it still can't be wrong to say that if that
  particular person is drinking, then everyone in the pub is drinking —
  because that person is, in fact, not drinking.

I can agree with the first case of the proof. But how is the second case true ? How can they apply material implication in the second case when the material conditional itself has not been proved yet or given to us ?

I would start at another way, 
Start with an other but similar problem:
Is the statement "If the last person has left the room then everybody has left the room" true?
(who would disagree? )
So "there is some person and when that person has left the room then everybody has left the room" 
and then replace "leaving the room" with "drinking"
and you are done
