[EtDybNuvCu]

Check out Metamath; we can do this for any serious foundations: http://us.metamath.org/downloads/metamath.pdf

[pacala]

Have somebody done it for the more general foundations based on category theory? If I understand correctly, Metamath is a set of tools enabling such a formulation, but not the formulation itself. See ML vs. theorem prover implemented in ML.

[mbid]

There are some developments, for example "Globular": https://arxiv.org/abs/1612.01093

I don't think there is a proof assistant that's really based on categorical foundations. I'd love to see something like that though.