It does not matter whether it is state-to-federal or federal-to-state, the same Fifth Amendment protection is applied in use and derivative use immunity, regardless of the source of the immunity. (
Murphy v. Waterfront Comm'n - 378 U.S. 52 (1964)).
At any rate, I would imagine that Obama might not want to create a full-blown constitutional crisis and loose the support of his party in Congress. History has shown that members of Congress support the power and authority of their branch of government when challenged by the executive branch.
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I want to hear Cunningham testify!