Prove/Disprove a claim in logic

Question

Prove/Disprove: $A, B$ are two formulas without common variables (meaning, $p$ is a variable of $A$ iff $p$ isn't variable of $B$, and vice-versa) and $\vDash A\to B$. Then, at least one of the following is true: $\vDash \lnot A$, $\vDash B$.

Now, as far as I understand $\vDash A\to B$ means that $A\to B$ is a tautology. Hence, it must be one of three:

• $A=t, B=t$
• $A=f, B=f$
• $A=f, B=f$

So clearly $\lnot A$ can't be a tautology and so is $B$.

Am I getting it the wrong way?

Answer 1

at least one of the following is true: $\vDash \lnot A, \vDash B$.

Assume not: i.e. $\nvDash \lnot A$ and $\nvDash B$.

This means that there is a valuation $v_1$ such that $v_1(A)=$t and a valuation $v_2$ such that $v_2(B)=$f.

But the two formulae have no common propositional variables; thus we can consider the valuation $v=v_1 \cup v_2$ and we have:

$v(A \to B)=$f

contradicting the assumption: $\vDash A \to B$.