Let A and B be any propositional formula

Question

How can I prove the following?

If $A$ is a tautology and $A\implies B$ is a tautology, then $B$ is a tautology.

Answer 1

Suppose $A$ is a tautology and $A \Rightarrow B$ is a tautology. Is there a truth-value assignment $v$ that would make $B$ false ? If so, then, since $A \Rightarrow B$ is a tautology, for this $v$ we would have that $A$ should be false. But then we have found a truth-value assignment for which $A$ is false. This contradicts the assumption that $A$ is a tautology.

Answer 2

$A$ is a tautology means that $A$ is always true.

So if $A$ is true and $A\implies B$ is true, then $B$ is always true.

So $B$ is a tautology.