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Universal Quantification in Knowledge

Proposition: For proving with knowledge K union {forall x: F} the goal G, it suffices to prove G with additional knowledge F[x <- T] for any term T:

K union {forall x: F}

G

~~>

K union {forall x: F, F[x <- T]}

G

"We have to prove G. Since we know (forall x: F), we have F[x <- T] and thus ...(proof of G with additional knowledge F[x <- T])."

A formula (forall x: F) in the knowledge is a machine that takes any T and produces additional knowledge F[x <- T].