Hamming code of a hamming code

Hamming code of a hamming code

For numbers of 3-bits we want to create a code that detect two bit errors.
The following solution is proposed:
Take a 3 bit number for example 000, codify with hamming code TWO times:
000 -> 000000 -> 0000000000
If we add 2 errors to the final code, 1100000000, hamming code correction
tell us that there is an error in the third position, so the new (invalid)
message is 1110000000, if we ignore the parity terms we obtain the code
100000, hamming code correction tell us that there is an error in the
first position, i.e. the new (valid) message is 000000 and if we ignore
again parity terms we obtain again 000.
I proved with many examples that this method works. If works, why works? I
can't find a counterexample where the method fails.