# No truth function that is expressed by a formula that uses only implication and equivalence connectives

I proved the following statement by induction:

Let $A$ be a propositional formula which uses only the connectives $→$ and $↔$. Prove (by induction on the complexity of $A$) that if every propositional variable is true then $A$ is true.

which was fairly straight forward to prove but then there was a follow up question which I was a little unsure of that referred to the previous proof:

Then deduce that there is a truth function $f$ that is not expressed by any propositional formula that uses only the connectives $→$ and $↔$

By the previous proof it was obvious to see that A would be true if every variable was true but if there was some variable that was not true it may be that A is not true. So by this the truth function may be different from the truth table of some propositional formula that uses only the connectives → and ↔ and then it would not be expressed by this formula?

Does this prove the statement correctly? My understanding of the truth functions and the idea of 'expressing a formula' So I'm unsure if my example is entirely true.