There is no need to worry, there are no ambiguities in the formal language
If we start with a formula that (as we know from the first derivation) is equivalent to and start rewriting it while preserving equivalence, then obviously all resulting formulas will be equivalent to . If, however, at some point we rewrite some A, which is equivalent to , to B such that but , then , but . In your case, you weakened to True.
This is the same as when we want to prove . If we rewrite the left-hand side exactly, then we can make it equal to the right-hand side, but if we drop , we can only prove and not the other way around.
In fact, dropping a conjunct is appropriate only when the conjunction is the topmost connective and when you prove implication, not equivalence. Otherwise, sometimes you get a weaker formula (e.g., ) and sometimes a stronger one (e.g., ). This is similar to how but when .