I would not say that they are equivalent but I would agree that the 3rd law is the key to understanding why momentum is conserved as it is.
The standard statement of Newton's 3rd Law is: " For every action, there is an equal and opposite reaction ."
So, if 2 particles collide, Newton's 3rd Law tells us that the force acting on each during the contact will be equal in magnitude and opposite in direction.
If we next invoke Newton's 2nd Law, often seen as
And for constant mass:
Stating the obvious, the particles will be in contact for the same amount of time . They have to be.
Therefore, for particles labelled as 1 and 2, and using the 2nd and 3rd Laws:
If we integrate both side of this wrt time, we have: