An antibacterial 7beta-(carboxyalkenoyl)amino-3-cephem-4-carboxylic acid represented by the following formula:
(wherein R is aryl or a heterocyclic group;
R' is hydrogen or halogen;
R2 is a single bond, alkylene, oxaalkylene, or thiaalkylene;
R3 is a hydrogen atom or carboxy modifying group;
R4 is hydrogen or methoxy;
R5 is hydrogen or a 3-substituent of cephalosporins;
R8 is a hydrogen atom or carboxy modifying group; and
X is oxygen, sulfur, or sulfinyl)
a pharmaceutical composition containing the same, and a method for treating a bacterial infection with the same.