At phase boundary given by $T(P)$ or $P(T)$, or $f(P,T) = 0$, we have $G_1(P,T) = G_2 (P,T)$.

Since $G_i = V_i \, \mathrm{d}P - S_i \, \mathrm{d}T$, we have $\Delta G = \Delta V \mathrm{d}P - \Delta S \, \mathrm{d}T = 0$.

Thus

$$ \frac{\mathrm{d}P}{\mathrm{d}T} = \frac{\Delta S}{\Delta V} = \frac{\Delta H}{T\Delta V} \quad . $$