7.3 Derived rules

In some books or tutorials more rules are allowed (apart from the basic 9) in order to deal with formulas more easily. They represent an abstraction: stop working in the details to dedicate our work in more complex problems (it's like the high level programming languages).

If you decide to use them, you will lose a lot of interesting work to do, but you will finish faster. My advice is to only use a rule if you know how to prove its validity by using the 9 basic rules.

Some of the ones I found at several places are:

• Law of double negation: allows changing $A$ to $\neg\neg A$ and viceversa.
• Modus Tollens: having $A\Rightarrow B$ and $\neg B$, then $\neg A$.
• Disjunctive syllogism: if $A\vee B$ and $\neg A$, then $B$. And if $A\vee B$ and $\neg B$, then it's $A$.
• Elimination of $\neg$$\Rightarrow$: if you have $\neg(A\Rightarrow B)$, then happen both $A$ and $\neg B$.
• Elimination of $\neg$$\vee$: if you have $\neg(A\vee B)$, then $\neg A$, and also $\neg B$.
• Elimination of $\neg\wedge$: if you have $\neg(A\wedge B)$, then $\neg A\vee\neg B$.
• Theorems which you can use when you want: $A\Rightarrow A$, $A\vee\neg A$, $\neg(A\wedge\neg A)$ and more.
• Change of equivalent formulas: if $A\Longleftrightarrow B$, then where it says $A$ you can put $B$ and viceversa.

There are lots more; but if someone requests you an exercise, they will tell you which rules are allowed and which not (for instance, in class we were allowed to use only the basic ones).