Prove an inequality of Prime Numbers The problem on which I am currently stuck is, 

Is it true that, $$x+y< \dfrac{p_{\pi(x)}+p_{\pi(y)}+p_{\pi(x)+1}+p_{\pi(y)+1}-2}{2}$$ for all sufficiently large $x$ and $y$, $x+y$ is a prime and $\pi(x)$ is the prime counting function?

My Atempt
I tried to prove the problem by showing that $$x<\dfrac{p_{\pi(x)}+p_{\pi(x)+1}-1}{2}$$and $$y<\dfrac{p_{\pi(y)}+p_{\pi(y)+1}-1}{2}$$
But I cannot prove or disprove it by any elementary means. Plotting graphs gives me the impression that both inequalities are true for sufficiently large $x$ but the process is very messy. 
So, can anyone give a simpler proof?
 A: The inequality seems extremely unlikely to be true for all large $x$ and $y$.  In particular, it seems likely you can always find large primes $p$ and $q$ such that $p+q-3$ is also prime.  If so, then let $x=p-2$ and $y=q-1$, in which case $p_{\pi(x)+1}=p$ and $p_{\pi(y)+1}=q$.  Noting that $p_{\pi(u)}\le p_{\pi(u)+1}-2$ in general, we have
$$
{p_{\pi(x)}+p_{\pi(y)}+p_{\pi(x)+1}+p_{\pi(y)+1}-2\over2}
\le{(p-2)+(q-2)+p+q-2\over2}=p+q-3=x+y
$$
In particular, if $2p-3$ is prime for infinitely many primes $p$, then the OP's inequality is violated infinitely often.  
I'm not aware of any argument for the existence of increasingly large primes $p$ and $q$ with $p+q-3$ also prime that doesn't invoke a conjecture.  If someone can think of one, I hope they'll post it.
A: It is not true for infinitely many pairs of $(x,y)$ with both $x,y$ arbitrarily large when we assume the Goldbach Conjecture. 
Then, for any odd prime $r$, we have by the GC that $r+3$ can be written as $p_m+p_n$. 
Therefore $r=p_m+p_n-3$, so take $x=p_m-1$, $y=p_n-2$. 
Then \begin{align*}
p_{\pi(x)}+p_{\pi(y)} &= p_{\pi(p_m-1)}+p_{\pi(p_n-2)} \\ &= p_{\pi(p_m-1)}+p_{\pi(p_n-2)} \\ &= p_{m-1} + p_{n-1} \\ &\leq p_m+p_n-4 \\ &= x+y-1 \end{align*}
On the other hand, $p_{\pi(x)+1}+p_{\pi(y)+1} = x+y+3$.
So:
$$\dfrac{p_{\pi(x)}+p_{\pi(y)}+p_{\pi(x)+1}+p_{\pi(y)+1}-2}{2}\leq x+y$$
Therefore it is, assuming the GC, not true for all sufficiently large $x,y$. 
