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For example, many systems have as an axiom the sentence "If P implies Q and P is the case, then Q is the case." To go along with the axioms the system will have a special rule of derivation, called the 'rule of substitution'.

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It says that you can derive from any axiom a sentence that is just like it, except that other sentences have been substituted for the 'P' and the 'Q'.

For example, from the axiom above, we can conclude the following: "If R&S implies that T or U, and R&S is the case; then it is the case that T or U." (This assumes that "R&S" and "T or U" are expressions in the formal system.) Most formal systems have either a rich set of rules of derivation, but few or no axioms; or a rich set of axioms but only the derivation rule of substitution.

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