Let $\phi: S \to \bar{S}$ be a diffeomorphism between two surfaces in $\mathbb{R^3}$. Such a map is called conformal if for all $p \in S$, and $v_1, v_2 \in T_p(S)$ (the tangent plane) we have
$$\langle d\phi_p(v_1), d\phi_p(v_2) \rangle = \lambda^2 \langle v_1, v_2 \rangle_p$$
for some nowhere-zero function $\lambda$.
$\phi$ is said to be angle-preserving, if
$$\cos(v_1, v_2) = \cos(d\phi_p(v_1), d\phi_p(v_2)),$$
which I take to mean
$$\frac{\langle v_1, v_2\rangle}{\lVert v_1 \rVert \lVert v_2 \rVert} = \frac{\langle d\phi(v_1), d\phi(v_2)\rangle}{\lVert d\phi(v_1) \rVert \lVert d\phi(v_2) \rVert} $$
From do Carmo, "Differential Geometry of Curves and Surfaces", 4.2/14:
Prove that $\phi$ is locally conformal if and only if it preserves angles.
The "only if" part is obvious, but how can the "if" portion be proved (i.e. how does preserving angles imply conformality)?