Period-doubling bifurcation theorem

Question

I have the following definition of period-doubling bifurcation for a one-parameter family of functions $\left\{{ F_{\lambda} }\right\}$ :

Definition. A family of functions $\left\{{ F_{\lambda} }\right\}$ undergoes a period-doubling bifurcation at the parameter value $\lambda=\lambda_{0}$ if there exist an open interval $I$ and an $\epsilon >0$ such that:

1. For each $\lambda \in [\lambda_{0}-\epsilon,\lambda_{0}+\epsilon]$, there exist a unique fixed point $p_{\lambda}$ for $F_{\lambda}$ in $I$.

2. For $\lambda \in (\lambda_{0}-\epsilon, \lambda_{0})$ , $F_{\lambda}$ has not cycles of period $2$ in $I$ and $p_{\lambda}$ is attracting(resp. repelling).

3. For $\lambda \in (\lambda_{0},\lambda_{0}+\epsilon)$, there exist a unique 2-cycle $q_{\lambda}^1,q_{\lambda}^2$ in $I$ with $F_{\lambda}(q_{\lambda}^1)=q_{\lambda}^2$. This 2-cycle is attracting (resp., repelling). Meanwhile, the fixed point $p_{\lambda}$ is repelling(resp., attracting).

4. As $\lambda \rightarrow{} \lambda_{0}$, $q_{\lambda}^i \rightarrow p_{\lambda_{0}}$.

Here, we use the notation $f^2=f \circ f$, and if the above conditions are valid with the order of the intervals $(\lambda_{0}-\epsilon, \lambda_{0})$, $(\lambda_{0},\lambda_{0}+\epsilon)$ switched , we also say that the family have a period-doubling bifurcation.

Once said that, I have found the following theorem which apparently implies the existence of a period-doubling bifurcation:

Theorem. Suppose that:

1. $f_{\lambda}(x_{0})=x_{0}$ for all $\lambda$ in an interval about $\lambda_{0}$.

2. $f'_{\lambda_{0}}(x_{0})=-1$.

3. $ \displaystyle \frac{\partial (f^2_{\lambda})'}{\partial \lambda} \Big |_{\lambda=\lambda_{0}}^{}(x_{0}) \neq 0 $.

Then there is an interval $I$ about $x_{0}$ and a function $p:I \rightarrow{\mathbb{R}}$ such that

$$ f_{p(x)}(x) \neq x $$

But,

$$ f^{2}_{p(x)}=x $$

My question is: Why this theorem implies the existence of such a bifurcation? I don't see it.

I understand that the condition $f'_{\lambda_{0}}(x_{0})=-1$ implies the first condition of the definition , to prove that we can apply the implicit function theorem.

I also see that the the existence of such a function $p$ implies the existence of a $2-$cycle, however, I can´t understand why this implies the other conditions of the definition.

Can anyone explain me please?

In advance, thank you.