I was wondering if this proof was ok or if (as I suspect) I am wrong and if you could help point me in the right direction.

thanks for any help

Suppose that is an inconsistent set of sentences. For each let be the set obtained from removing G from .

Prove that for any .

Proof

We can split into two cases.

1. is still an inconsistent set and so it can prove anything, namely

2. is now consistent in which case G was false in the cases that was satisfied and so is true when is satisfied and so by definition and by completness