I want to prove that $\neg(A \Rightarrow B) \iff A \land \neg B$ holds without using a truth table.
"$\Leftarrow$": This one is simple: Suppose $A \land \neg B$. We want to show: $(A \Rightarrow B) \Rightarrow \bot$. For that we suppose $A\Rightarrow B$. Now our goal is $\bot$. Since by our assumption $A$ and $A\Rightarrow B$ are true we get $B$ by using Modus ponens. Since $B$ and $\neg B$ holds we get $\bot$ by using Modus ponens again. $\square$
How does "$\Rightarrow$" work?
Thanks in advance!