Let $v_0$ be the valuation that assigns true ($T$) to every propositional variable.
I'm trying to show that any formula $\phi$ is logically equivalent to one with only propositional variables and the binary connectives $\wedge$ and $\to$ if and only if the natural extension of $v_0$, $v$ say, assigns the value $T$ to $\phi$.
If $\phi$ can be expressed in such a way then clearly $v(\phi)=T$ from the truth tables of $\wedge$ and $\to$. Now I have trouble proving the other way around formally. I think I can believe it is true looking at different truth tables but can't put my thoughts in order.
Could you tell me how you would go at it so I can get my head around it? I would be very grateful!