short rate - Bond dynamics in Ho Lee model

The short rate in the Ho-Lee model is given by :

$$dr_t=\left( \frac{df(0,t)}{dt} +\sigma^2t\right)dt + \sigma dW_t$$

I'm trying to find the bond dynamics given by :

$$dP(t,T)/P(t,T)=r_tdt-\sigma(T-t)dW_t$$

I started from :

$$P(t,T)=E_t[e^{-\int_t^T r_sds}]$$

and I applied Itô to the function $P(t,T)=\phi(t,r)$:

$$d\phi(t,r) = \frac{\partial \phi(t,r)}{\partial t}dt+\frac{\partial \phi(t,r)}{\partial r} dr_t+ \frac{1}{2} \frac{\partial^2\phi(t,r)}{\partial r^2}(dr_t)^2$$

I computed the derivatives :

$$\frac{\partial \phi(t,r)}{\partial t}=r_tP(t,T)$$

$$\frac{\partial \phi(t,r)}{\partial r} = -(T-t)P(t,T)$$

$$\frac{1}{2} \frac{\partial^2\phi(t,r)}{\partial r^2} = (T-t)^2P(t,T)$$

Assembling everything I get :

$$dP(t,T)/P(t,T) = r_tdt-(T-t)\sigma dW_t +\left[ \frac{1}{2}(T-t)^2\sigma^2-(T-t)\left( \frac{df(0,t)}{dt}+\sigma^2t \right) \right] dt$$

I don't know how to get rid of the last $dt$ term. Any Help? Or did I get the derivatives wrong? I checked them several times but I don't see where the probem comes from. Thank you