Let  and  be quantifier-free formulas. It is a simple exercise to prove that the sentence  is logically equivalent to the sentence  Here, we generalize this simple fact, presenting various sentences that are logically equivalent to each sentence of the form  where  are quantifier-free formulas.

We prove the logical equivalence by the game method, which assigns each sentence y and structure M (interpreting each symbol in  to a two-players game  (the players are  and  such that the sentence  holds in M if and only if player  has a winning strategy in the game

truth value, semantics, first-order logic.