Primes between consecutive prime-index-primes

[cino hilliard]

Hi seq fans,
I submitted a sequence of primes that are between prime(prime(x)) and 
prime(prime(x+1).

This is similar to  A007821
           2,7,13,19,23,29,37,43,47,53,61,71,73,79,89,97,101,103,107,
           113,131,137,139,149,151,163,167,173,181,193,197,199,223,227,
           229,233,239,251,257,263,269,271,281,293,307,311,313,317,337,
           347,349,359,373
           Primes p(n) where n is composite.

Except 2 is not allowed in my sequence.

Conjecture: For x > 1 there is at least 1 prime between prime(prime(x) and
prime(prime(x+1).

Can someone prove this or provide a link?

Thank,
Cino

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