# Exchange principle in terms of states and coordinates?

I have seen the exchange principle written in two ways, one in terms of coordinates and the other in terms of states:

If $\psi_{AB}(1,2)$ represents particle $A$ in state $1$ and particle $B$ in state $2$ then for bosons: $$\psi_{AB}(1,2)=\psi_{AB}(2,1)$$ and for fermions: $$\psi_{AB}(1,2)=-\psi_{AB}(2,1)$$

and

If $\psi_{AB}(\vec x_1, \vec x_2)$ represents particle $A$ at $\vec x_1$ and particle $B$ at $\vec x_2$ then for bosons: $$\psi_{AB}(\vec x_1,\vec x_2)=\psi_{AB}(\vec x_2,\vec x_1)$$ and for fermions: $$\psi_{AB}(\vec x_1,\vec x_2)=-\psi_{AB}(\vec x_2,\vec x_1)$$

Do both these tell us the exactly same information (i.e. may one hold when the other doesn't) and can this be shown?

I am pretty sure that the vector $\vec{x}$ includes spin coordinates. The coordinate itself is an observable which has a continuum spectrum. So a coordinate operator has eigenvalues that characterize a state. So there is no much difference between the first and second definition, since instead of saying an electron has a coordinate $x$, we can also say that the electron is in the state $x$.