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Proofs
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A proof of a conclusion from a set of premises is a sequence of sentences terminating in the conclusion in which each item is either
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a premise,
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an instance of an axiom schema, or
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the result of applying a rule of inference to earlier items in sequence
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Example:
1. p → q
Premise 2. q → r
Premise 3. ( q → r ) → ( p → ( q → r ))
II 4. p → ( q → r )
MP : 3,2 5. ( p → ( q → r )) → (( p → q ) →( p → r ))
ID 6. ( p → q ) → ( p → r )
MP : 5,4 7. p → r
MP : 6,1