 That this argument is valid is perhaps most easily seen by a reductio argument, in which one assumes that the  conclusion—(7)—is false, and then shows that the denial of (7), along with premises (1) through (6), leads to a contradiction.  Thus if, contrary to (7), God exists, it follows from (1) that God is omnipotent, omniscient, and morally perfect.  This, together with (2), (3), and (4) then entails that God has the power to eliminate all evil, that God knows when evil exists, and that God has the desire to eliminate all evil.  But when (5) is conjoined with the reductio assumption that God exists, it then follows via modus ponens from (6) that either God doesn’t have the power to eliminate all evil, or doesn’t know when evil exists, or doesn’t have the desire to eliminate all evil.  Thus we have a contradiction, and so premises (1) through (6) do validly imply (7).