It is possible to begin from false assumptions, perform valid operations, and arrive at a true statement. For instance, assume $0=1$. Multiplying both sides by $0$ yields the true statement $0=0$.
For this reason, you cannot prove that a statement is true by assuming that it is true and then deriving something else that is true. This is a common mistake I've seen among calculus students. Often they just need to write their equations in a different order, and everything works out because the operations they're doing are reversible, but the logical order of statements is important.
If it was possible to derive a true result from a false premise using valid operations, then it would also be possible to work backwards from a correct result to a false premise by performing valid operations.
This statement is untrue exactly for the reason that not every valid operation is reversible. For instance, in my example we cannot reverse multiplication by $0$ by dividing by $0$. It's not a valid operation. If you're familiar with dot products, then you know that dotting by a vector is not reversible - i.e., if we know $u \cdot w = v\cdot w$, we cannot undo dotting with $w$ to obtain $u=v$. There are tons more examples of operations we do that are not reversible, but this should give you the idea.