I am trying to understand the properties of a proton-neutron system assuming that isospin is a good symmetry, so I will forget about electromagnetism, weak interactions, QCD and all those more sophisticated ideas. Let's consider then that the nucleon is a fundamental particle with isospin $T=1/2$, so that $T_3=1/2$ corresponds to a proton and $T_3=-1/2$ corresponds to a neutron. My question is, in this situation, can the proton and the neutron be considered as indistinguishable fermions, and therefore any state of two of these nucleons must be antisymmetric with respect to the interchange of the dynamical variables corresponding to each nucleon? In general, that would mean that a state where $N$ and $N'$ represent any two nucleons (i.e., any values for the third component of isospin) satisfies:
$$ \Phi_{\xi,N;\,\xi',N'} = - \Phi_{\xi',N';\,\xi,N} $$
where by $\Phi_{\xi,N;\,\xi',N'}$ I mean a state in the Hilbert space of the theory containing a nucleon of type $N$ with other variables $\xi$ (momentum, third component of spin, ...) and another nucleon of type $N'$ with other variables $\xi'$. I know that this must be true if $N=N'$, e.g., for two protons we must have:
$$ \Phi_{\xi,p;\,\xi',p} = - \Phi_{\xi',p;\,\xi,p} $$
but the really interesting case for me is the one where we have a proton and a neutron:
$$ \Phi_{\xi,p;\,\xi',n} = - \Phi_{\xi',n;\,\xi,p} $$
If this were true, that would mean that the proton and the neutron are in fact indistinguishable, assuming of course that isospin is a good symmetry (which I know it isn't, but the point here is to understand how far the idea of indistinguhishability can be taken in a theory with internal symmetries such as isospin).
