We have two statements:

1. 0 = 1
2. Both statements are false

If the second statement is true, then it is false. We've come to a contradiction.

If the second statement is false, then at least one of the statements is true. The second one is false, therefore, the first one is true.
We've proven that 0 = 1.

How is that possible and is there a way to prove this is wrong?