To enhance the user's privacy in electronic ID, anonymous credential systems have been researched. In the anonymous credential system, a trusted issuing organization first issues a certificate certifying the user's attributes to a user. Then, in addition to the possession of the certificate, the user can anonymously prove only the necessary attributes. Previously, an anonymous credential system was proposed, where CNF (Conjunctive Normal Form) formulas on attributes can be proved. The advantage is that the attribute proof in the authentication has the constant size for the number of attributes that the user owns and the size of the proved formula. Thus, various expressive logical relations on attributes can be efficiently verified. However, the previous system has a limitation: The proved CNF formulas cannot include any negation. Therefore, in this paper, we propose an anonymous credential system with constant-size attribute proofs such that the user can prove CNF formulas with negations. For the proposed system, we extend the previous accumulator for the limited CNF formulas to verify CNF formulas with negations.