The ($^{11}$B,$^{11}$Li) double charge-exchange reaction (DCER) at $E(^{11}$B)/$A$=80 MeV was measured for the first time to demonstrate the feasibility of the reaction for studying neutrino nuclear responses for double beta decays (DBD). The $^{13}$C($^{11}$B,$^{11}$Li)$^{13}$O reaction shows strengths at the ground state and low and high excitation giant resonance regions. The $^{56}$Fe ($^{11}$B,$^{11}$Li) $^{56}$Ni reaction shows the large strengths in the possible double giant resonance region and beyond, but shows no strengths in the low excitation region below 5 MeV, suggesting strong concentration of the DBD strength at the high excitation region. The DCER is used to evaluate the spin isospin strengths relevant to DBD responses.
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