For a joint life annuity on lives (x) and (y) with payments of 1 per year in advance, the EPV can be written recursively as:
begin{eqnarray*}
ddot{a}_{xy} = 1 + v p_{xy} ~ddot{a}_{x+1:y+1}.
end{eqnarray*}
The EPV is
begin{eqnarray*}
ddot{a}_{xy} = sum_{k=0}^{infty} v^k ~_k p_{xy} = ddot{a}_{xy}^{00}.
end{eqnarray*}
By conditioning (or using the Chapman-Kolmogorov equations), we may write
begin{eqnarray*}
~_{k+1} p_{xy} = ~_{k+1} p_{xy}^{00} = p_{xy}^{00} ~_k p_{x+1:y+1}^{00} = p_{xy} ~_k p_{x+1:y+1} .
end{eqnarray*}
Thus, using (k=j+1), we have
begin{eqnarray*}
ddot{a}_{xy} &=& 1 + sum_{k=1}^{infty} v^k ~_k p_{xy} = 1 + sum_{j=0}^{infty} v^{j+1} ~_{j+1} p_{xy} \
&=& 1 + v p_{xy} sum_{j=0}^{infty} v^j ~_j p_{x+1:y+1} =1 + v p_{xy} ddot{a}_{x+1:y+1}.
end{eqnarray*}