Assuming p V !p (read: p or not p) as an axiom means that you are admitting excluded middle. This axiom is part of "classical propositional logic". So a proof u

Assuming p V !p (read: p or not p) as an axiom means that you are admitting excluded middle. This axiom is part of "classical propositional logic". So a proof using this case distinction (p true or p false) would be valid, in the classical sense. There is a stronger system, called "intuitionistic propositional logic", where this axiom is not valid. There are less formulae that are valid intuitionistically, but any formula that is valid intuitionistically is also valid clasically. There are various philosophical reasons why one prefers intuitionistic logic, but note that in any case it is stronger to have an intuitionistic proof, so these are preferable (when they exist).