A logical operator is said to be truth functional when the truth value of sentences in which it is the major operator is dependent only on the truth values of component atomic sentences. For example, the negation operator is truth functional because the truth value of $\neg A$ is dependent only on the truth value $A$.
How do you show that the hypothetical logical operator $ H(X) \leftrightarrow $ "It is necessary that $X$" is not truth functional?
In my attempt, I provide 2 distinct true sentences $ A $ and $ B $ such that $H(A)$ is true but $H(B)$ is false. Namely, A = "1 + 1 = 2" and B = "Humans exist." Because the truth or falsity of $H(X)$ is not fixed when $X$ is true, it cannot be said to be dependent on the truth value of $X$ alone.
Is this proof satisfactory? Also, is there a more rigorous proof?