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 ?

"If 1+1=3, then I am the King of France" is an example of a "vacuously true" statement - in classical logic, we take "P -> Q" as equivalent to "Q or !P". In this case "!P", so "Q or !P" is true (regardless of the truth value of Q), and so "P -> Q" is true; although this tells us little we didn't know before!
Similarly, in the second case you note, there exists a person A for whom "A is drinking" is false. Therefore we can take "A is drinking -> everyone is drinking" as equivalent to "(everyone is drinking) or !(A is drinking)" which is true, since "!(A is drinking)" is true regardless of whether (everyone is drinking) is true or not.
