Perfectly secure shift cipher Prove that if only one character is encrypted using a shift cipher, then the shift cipher is perfectly secure.

I want to show that $P(P=p | C=c)=P(P=p)$. But I don't know how to relate. Can anyone guide me?
Since you encrypt just a single letter, there are $26^2$ combinations of $p$ and $c$ where $c=E(p)$. This is because there are $26$ possible shift keys in the key space, an therefore each $p$ can be mapped to one of $26$ letters in the code space. Now, assuming that the key is distributed uniformly in the key space, each of those combinations of $(p,c)$ has a probability $\frac{1}{26^2}$. From base low we have:
 $$P(P=p\mid C=c) = \frac{P(P=p\text{ AND }C=c)}{P(C=c)}.$$ Now,
 $$P(P=p\text{ AND } C=c) = P(p,c) = \frac{1}{26^2},$$ and assuming uniform distribution $P(C=c) = 1/26$, you get $P(P=p|C=c) = \frac{1}{26} = P(P=p)$. QED