Relation between erfc and the BER
Assuming your bit error probability formulas are correct, for antipodal signalling

$$P_{b, aS} = \frac{1}{2} \mathrm{erfc}\left(\sqrt{\frac{E_{s,aS}}{N_0}}\right)$$

and for on-off signalling

$$P_{b, oS} = \frac{1}{2} \mathrm{erfc}\left(\sqrt{\frac{E_{s,oS}}{2N_0}}\right)$$

If these two error probability are identical, i.e. $P_{b, aS} = P_{b, oS}$,

$$\frac{1}{2} \mathrm{erfc}\left(\sqrt{\frac{E_{s,aS}}{N_0}}\right) = \frac{1}{2} \mathrm{erfc}\left(\sqrt{\frac{E_{s,oS}}{2N_0}}\right) \tag{1}$$

Because both $\mathrm{erfc(.)}$ and $\sqrt{.}$ are bijective functions, $(1)$ implies $\frac{E_{s,oS}}{N_0} = 2\frac{E_{s,aS}}{N_0}$ in linear scale and 3dB difference in log scale.