Catalysis of the reaction: 2-(formamido)-N(1)-(5-phospho-D-ribosyl)acetamidine + ATP = 5-amino-1-(5-phospho-D-ribosyl)imidazole + ADP + 2 H+ + phosphate.
2-(formamido)-1-N-(5-phosphoribosyl)acetamidine cyclo-ligase (ADP-forming)