The Collaborative International Dictionary
Antiptosis \An`tip*to"sis\, n. [L., fr. Gr. ?; ? against + ? a falling, a case, ? to fall.] (Gram.) The putting of one case for another.
Wiktionary
n. (context grammar rhetoric English) Substitution of one grammatical case for another.
Wikipedia
Antiptosis, which translates from the Greek ανταλλαγή (exchange of) and περίπτωση (case), is a rhetorical device. Specifically, it is a type of enallage (the substitution of grammatically different but semantically equivalent constructions) in which one grammatical case is substituted for another.
In English, this technique is utilized only with pronouns, and is more effective with languages that utilize inflected nouns, such as Greek and Latin (or any other derivative romance language).