Basis Properties
Since all axioms and rules of the calculus are schemata, derivation is closed under substitution:
- If then
where σ is any substitution (of formulas using only implication).
The implicational propositional calculus also satisfies the deduction theorem:
- If, then
As explained in the deduction theorem article, this holds for any axiomatic extension of the system consisting of axioms 1 and 2 above and modus ponens.
Read more about this topic: Implicational Propositional Calculus
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