"Proof" that "ZFC + there exists an inaccessible cardinal" is consistent

Question

I have a proof that this theory is consistent using this theory itself. I want to know what's wrong with this proof:

1. "ZFC + there exists an inaccessible caridnal" proves that "ZFC is consistent".

2. "There exists an inaccessible cardinal" is a statement that is independent from ZFC.

3. Combining 1 and 2, we have "ZFC + there exists an inaccessible cardinal" is consistent.

I'm able to conclude (3) because we proved ZFC is consistent in (1) and also that "There exists an inaccesible" is independent from ZFC. If $X$ is a consistent theory and $P$ is a statement independent from $X$, then both $P+X$ and $P+\neg X$ are consistent theories.

What's wrong with this proof?