Final answer:
The deep vein of the anterior abdominal wall in question is the inferior epigastric vein, which ultimately drains into the external iliac vein.
Step-by-step explanation:
The inferior epigastric vein is the deep vein of the anterior abdominal wall that you are inquiring about. It collects blood from the lower anterior abdominal wall and ultimately drains into the external iliac vein. The inferior epigastric vein is not as large or as widely known as some other veins in the abdominal region, such as the inferior vena cava which is the major vein that carries most of the blood from the lower body back to the heart, but it plays a crucial role in the venous drainage of the abdominal wall.