Proof that $\binom{2\phi(r)}{\phi(r)+1} \geq 2^{\phi(r)}$ I try to prove the following
$$\binom{2\phi(r)}{\phi(r)+1} \geq 2^{\phi(r)}$$
with $r \geq 3$ and $r \in \mathbb{P}$. Do I have to make in induction over $r$ or any better ideas?
Any help is appreciated.

Combinatorial proof of ${2n \choose n+1} \geq 2^n$ where $n \geq 2$:
Let's take set $\{x_1,y_1,\dots,x_{n-2},y_{n-2},a,b,c,d\}$ which has $2n$ elements; select three elements out of $\{a,b,c,d\}$ and for all $i$, a single element of $\{x_i,y_i\}$, you'll select $n+1$ in total. So
${2n \choose n+1} \geq {4 \choose 3} 2^{n-2}=2^n$
