If you try to simplify it to an equation, you end up with

$is_valid = ($a ? ($d ? $b && $c : !$b && !$c || !$b && $c || $b && !$
+c) : !($b && $c && $d);
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which is totally unreadable and unmaintainable, doubly so when you use the full variable names. So don't do this :)