Friedman has made public on his website a 2016 draft titled "Expanding Mind Theory". On page 4 there is a definition of a theory $\mathrm{EM}$ in first-order logic which formalizes the theory of two minds.
Neither of the bold statements have proofs appearing in this document. There are two theorems working toward calibrating the logical strength of $\mathrm{EM}$ as in the first bold statement: theorem 2.2, that $\mathrm{ZF}$+"there exists a $1$-extendible cardinal" proves consistency of $\mathrm{EM}$, and theorem 3.2.19, that $\mathrm{ZFC}$ is interpretable in $\mathrm{EM}$. The final section hints toward the existence of strengthened versions of this theory as in second bold statement, except with still two minds rather than infinitely many.