  It is known that one can construct non-parametric functions by assuming
classical axioms. Our work is a converse to that: we prove classical axioms in
dependent type theory assuming specific instances of non-parametricity. We also
address the interaction between classical axioms and the existence of
automorphisms of a type universe. We work over intensional Martin-L\"of
dependent type theory, and in some results assume further principles including
function extensionality, propositional extensionality, propositional
truncation, and the univalence axiom.
