Error detection/correction algorithm We have 2 stations that communicate with each other, but we need to detect (or even correct) when something is wrong.

We use 8 binary words: each consisting of 3 bits  and to send it we code it as 
  where  is the complement of  and  is the even parity check bit of  .

We need to find the capabilities of this code (up to how many can we detect and how many can we correct). BUT, a proof is required.

This is how far I've reached so far:

First we find the hamming distance:
If  changes then  changes, also  changes. So we have a hamming distance of 3.

This means that we can detect two bit errors or correct a single error. 

Can you help me write the proof for that?

(also the tags may need some refinement - comments about the downvote are welcome)
Consider any two codewords based on $b_2b_1b_0$ and $c_2c_1c_0$. The Hamming distance between the two codewords is at least twice the Hamming distance between $b_2b_1b_0$ and $c_2c_1c_0$ (why?). Furthermore, if the Hamming distance between $b_2b_1b_0$ and $c_2c_1c_0$ is exactly $1$, then your argument shows that the Hamming distance between the codewords is $3$. We can conclude that the Hamming distance is always at least $3$.