Note: in the following rules, the greek letters "φ" and "ψ" are names of SL wffs.

**A (Rule of Assumption)**

You can write down any SL wff, depending on itself.

**&I (Conjunction Introduction)**

If you have derived φ and ψ, you can write down (φ&ψ), depending on everything φ and ψ depend on.

**&E (Conjunction Elimination)**

If you have derived (φ&ψ),
you can write down φ or ψ,
depending on everything (φ&ψ) depends on.

**→I (Conditional Introduction)**

If you have assumed φ, and you have derived ψ,
you can write down (φ→ψ), depending on everything ψ depends on except φ.

**→E (Conditional Elimination or Modus Ponens)**

If you have derived (φ→ψ) and φ, you can write down ψ, depending on everything (φ→ψ) and φ depend on.

**~I (Negation Introduction)**

If you have assumed ψ, and you have derived (φ&~φ), then you can write down ~ψ, depending on everything (φ&~φ) depends on except ψ.

**~E (Negation Elimination)**

If you have assumed ~ψ, and you have derived (φ&~φ), then you can write down ψ, depending on everything (φ&~φ) depends on except ~ψ.

**∨I (Disjunction Introduction)**

If you have derived φ, you can write down (φ∨ψ)
or (ψ∨φ), depending on everything φ depends on. (ψ is any SL wff.)

**∨E (Disjunction Elimination or Disjunctive Syllogism)**

If you have derived (φ∨ψ) and ~ψ,you can write down φ,
depending on everything (φ∨ψ) and ~ψ depend on.

Also, if you have derived (φ∨ψ) and ~φ, you can write down ψ, depending on everything (φ∨ψ) and ~φ depend on.

**PC (Proof by Cases)**

If you have derived (φ∨ψ) and (φ→α)and (ψ→β), then you can write down (α
∨β),depending on everything (φ∨ψ) and (φ→α) and (ψ→β) depend on.

**↔I (Biconditional Introduction)**

If you have derived ((φ→ψ)&(ψ→φ)),
you can write down (φ↔ψ), depending on everything ((φ→ψ)&(ψ→φ)) depends on.

**↔E (Biconditional Elimination)**

If you have derived (φ↔ψ), you can write down ((φ→ψ)&(ψ→φ)) depending on everything (φ
↔ψ) depends on.