Propositional Logic: Entailment
It's not easy to give a full explanation without knowing your axioms and rules, however, $A \land \neg A$ is a contradiction, and it is by exploiting that, that you can prove whatever you like.
First, let $\top = A \lor \neg A$.
We know that &quot;true or whatever else&quot; must be true.  Since $\top$ is a tautology, then $\top \lor B$ must be true.
But since $A \land \neg A = \neg (A \lor \neg A) = \neg \top$ is assumed to be true we conclude the following:

from $\top \lor B$ and $\neg \top$
$B$


It all hinges on the fact that $A \lor \neg A$ is a tautology and that you assume that the negation of a tautology is true.
So to answer your question &quot;actually how can $A$ and $\neg A$ occur at the same time?&quot; It cannot, unless you have a contradiction.  When you assume a contradiction, you can prove anything.