Meanwhile, a car bomb in the northern city of Aleppo killed 12 government soldiers, the activists said.
The attacks come as Syrian government forces press an offencive in the outskirts of the capital, and an 11-nation group that includes the US meets in the Qatari capital of Doha to coordinate military aid and other forms of assistance to the rebels.
The explosion in the capital went off behind a bakery in the Ruken al-Deen neighbourhood, according to the Britain-based Syrian Observatory for Human Rights. It reported 10 wounded and said there was no immediate claim of responsibility.
Later, three rebels attacked Ruken al-Deen's police station, the Observatory said. The attack left the three rebels and four policemen dead and nine others wounded. Meanwhile, the Observatory said rockets fell in the Damascus suburb Jarmana and the Abbasid district in the heart of the capital. It said there was material damage and unspecified number of injuries.
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The Syrian army has been moving against rebels in districts outside Damascus that are used as launching pads to attack the capital, President Bashar Assad's seat of power.
It provided no other details, but both the government and the opposition have recently declared offensives in Aleppo. Meanwhile, the group dubbed the Friends of Syria have been meeting in Qatar. Yesterday, the donors agreed to do more to help the embattled rebels, US Secretary of State John Kerry said.