Show that law of conservation of linear momentum and Newton's third law are the same

According to the third law of motion, Action reaction forces are equal and the total external force on the system being zero , momentum is conserved.

F1 = F2
ma1 = ma2
mdv1/dt = mdv2/dt
dmv1/dt = dmv2/dt
dp1/dt = dp2/dt
integrating with time dt gives,
p1 = p2

Consider a collision between two bodies. Since they exert equal and opposite forces on each other for the same time, they produce equal and opposite changes in momentum in each other--thus the total momentum of the system remains unchanged.