AND Intro
A
B
------
A ^ B
============
AND Elim
(A ^ B)
-------
A or B
============
OR Intro
(~A -> B)
---------
(A v B)
============
OR Elim
(A v B)
(A -> C)
(B -> C)
---------
C
============
IMPLY Intro
A
----
B -> A
============
IMPLY Elim
(A -> B)
A
--------
B
============
NOT Intro
(A -> B)
(A -> ~B)
---------
~A
============
NOT Elim
~~A
-----
A
============
& Conditional Proof, Universal Intro/Elim, Existential Intro/Elim, Identity Intro/Elim.