An antibacterial 7beta-(carboxyalkenoyl)amino-3-cephem-4-carboxylic acid represented by the following formula: ##STR1## (wherein R is aryl or a heterocyclic group; R.sup.1 is hydrogen or halogen; R.sup.2 is a single bond, alkylene, or thiaalkylene; R.sup.3 is a hydrogen atom or carboxy modifying group; R.sup.4 is hydrogen or methoxy; R.sup.5 is hydrogen or a 3-substituent of cephalosporins; R.sup.6 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.