We show that a stationary solution of the Einstein-Maxwell equations which isclose to a non-degenerate Reissner-Nordstr\"om-de Sitter solution is in factequal to a slowly rotating Kerr-Newman-de Sitter solution. The proof uses thenon-linear stability of the Kerr-Newman-de Sitter family of black holes forsmall angular momenta, recently established by the author, together with anextension argument for Killing vector fields. Our black hole uniqueness resultonly requires the solution to have high but finite regularity; in particular,we do not make any analyticity assumptions.
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