You might want to check the values of the arguments. For instance, ForAll[n, False, False] evaluates to True. So does ForAll[n, False, x], because from the hypothesis False, any consequent follows.
You should find that each of EvenQ[n], IntegerQ[i], IntegerQ[j] evaluates to False, because like most *Q functions EvenQ[] and IntegerQ[] evaluate immediately to False if the argument is an expression explicitly does not satisfy the condition being tested as is. For instance, n has the head Symbol, not Integer.
