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