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We prove a conservativity result for extensional type theories overpropositional ones, i.e. dependent type theories with propositional computationrules, using insights from homotopy type theory. The argument exploits a notionof canonical homotopy equivalence between contexts, and uses the notion of atype-category to phrase the semantics of theories of dependent types.Informally, our main result asserts that, for judgements essentially concerningh-sets, reasoning with extensional or propositional type theories isequivalent.

A conservativity result for homotopy elementary types in dependent type theory