Prenex normal form

A formula of the predicate calculus is in prenexnormal form if it is written as a string of quantifiers (referred to as the prefix) followed by a quantifier-free part (referred to as the matrix).

Every formula in classical logic is equivalent to a formula in prenex normal form. For example, if ${\displaystyle \phi (y)}$, ${\displaystyle \psi (z)}$, and ${\displaystyle \rho (x)}$ are quantifier-free formulas with the free variables shown then

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