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Proposition: For proving with knowledge K union {exists x: F} the goal G, it suffices to prove G with additional knowledge F[x <- a] for some object constant a that does not appear in K, G, or F:
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K union {exists x: F} G (a not in K, G, F)
K union {exists x: F, F[x <- a]} G
"We prove G. Since we know (exists x: F), we have have some a with F[x <- a]. Thus ...(proof of G with new knowledge F[x <- a])."
A formula (exists x: F) in the knowledge base is an "engine" which returns a new constant a about which we know (only) F[x <- a].