This is a handy reference to all of the rule of logical inference I could dig up. A fun guide for all you logicians.

Modus Ponens
If A, then B.
A.
Therefore B.

Modus Tollens
If A, then B.
Not B.
Therefore not A.

Hypothetical Syllogism
If A, then B.
If B, then C.
Therefore if A, then C.

Disjunctive Syllogism
P or Q.
Not P.
Therefore Q.

Conjunction
A.
B.
Therefore A and B.

Simplification
A and B.
Therefore A.

A.
Therefore A or B.

Here, they start to get more complex, so I will use some basic notation. X -> Y means if X, then Y. ~X means not X. :: means is the same as. iff is "if and only if", for any sentence if x iff y, then y iff x. It's a mutal dependency thing, dig? FORALL is the upsidedown "A", FORALLx means, guess what? For all members of x. FOR SOME is the backwards "E". FOR SOME x means there is at least one member of x that the statement to follow is true about.

Tautology
A :: (A or ~A)
A :: (A AND A)

Double Negation
~~A :: A

Transposition
A -> B :: ~B -> ~A

Material Implication
A -> B :: ~A or B

Material Equivalence
A iff B :: ((A -> B) AND (B -> A))
A iff B :: ((A AND B) or (~A AND ~B))

Constructive Dilemma
(A -> B) AND (C -> D)
A or C.
Therefore B or D.

Commutativity
(A or B) :: (B or A)
(A AND B) :: (B AND A)

Associativity
(A or (B or C)) :: ((A or B) or C)
(A AND (B AND C)) :: ((A AND B) or C)

Distribution
(A AND (B or C)) :: ((A AND B) or (A AND C)
(A or (B AND C)) :: ((A or B) AND (A or C)

Exportation
((A AND B) -> C) :: (A -> (B -> C))

Universal Instantation
FORALLx
All instances of x can be replaced by any name.

Existential Instantation.
FOR SOMEx
All instances of x can be replaced by a new name (cannot have been previously mentioned in a proof).