Proofs about Proofs

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\textbf{If A ⊢ B then ⊢ A→B}

Proof Given a proof for A ⊢ B …

… we can turn it into a proof for ⊢ A→B:

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\textbf{If ⊢ A→B then A ⊢ B}

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\textbf{If A ⊢ B then A ⊢ ¬¬B}

Proof:

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\textbf{If A ⊢ C then A ⊢ B→C}

\textbf{If A ⊢ B and A ⊢ ¬C then A ⊢ ¬(B→C)}