Boolean Algebra Simplifying complex equation
Whenever you are faced with two Boolean expressions $f,g$ on $n$ variables and wish to know whether they are equivalent, there is a simple algorithm you can apply:


  Go over all $2^n$ possible truth assignments, and check whether $f$ and $g$ have the same truth value on each.


While this is infeasible for large $n$, in your case $n = 4$, so there are only 16 many truth assignments to check; in fact, you could even do this by hand, though it might be less error-prone if you programmed it.

Using this approach, you should be able to tell whether your simplification is valid.