In[]:=
TrueQ[DiracDelta[x,y]==DiracDelta[y,x]]
Out[]=
True
In[]:=
TrueQ[FullSimplify[DiracDelta[x,y,z]==DiracDelta[x]DiracDelta[y]DiracDelta[z]]]
Out[]=
True
In[]:=
TrueQ[FullSimplify[DiracDelta[r-r0,theta-theta0,phi-phi0]==DiracDelta[r-r0]DiracDelta[theta-theta0]DiracDelta[phi-phi0]]]
Out[]=
True
In[]:=
TrueQ[FullSimplify[DiracDelta[r-r0,theta-theta0,phi-phi0]==DiracDelta[r-r0]DiracDelta[theta-theta0]DiracDelta[phi-phi0]/(r^2Sin[theta])]]
Out[]=
False