If $f$ i twice differentiable scalar function and $X_t, Y_t$ are Ito processes then Ito lemma holds. But in 90% of sources I can only find the case, when $Y_t=t$ (it is deterministic function). The Ito formula is given by:
$$df=(\mu f_x+f_t+\frac{\sigma^2}{2}f_{xx})dt+\sigma f_xdB_t.$$
But what if $Y_t$ is not deterministic function but "normal" Ito process?
If $f: \mathbb{R} \times \mathbb{R} \longrightarrow \mathbb{R}$ then $Z_t = f(X_t,Y_t)$ is a stochastic process given by $$dZ_t = f_x dX_t + f_ydY_t + \frac{1}{2}f_{xx}d[X]_t+ \frac{1}{2}f_{yy}d[Y]_t + f_{xy}d[X,Y]_t$$ where $d[X]_t$ is the quadratic variation of $X_t$ and $d[X,Y]_t$ is the quadratic covariation of $X_t$ and $Y_t$.
People often write informally $d[X]_t = ``(dX_t)^2"$ and $d[X,Y]_t = ``(dX_t)(dY_t)"$.
(So if $dX_t=adt+bdB_t$ and $dY_t=αdt+βdB_t$ then $d[X]_t = b^2dt$, $d[Y]_t = \beta^2 dt$ and $d[X,Y]_t = b\beta dt$)