5.3: Substituting into an argument

Substituting into an argument does not change its validity.

Theorem $\PageIndex{1}$: Substitution Rule

Suppose $A_1, A_2, \ldots, A_m \therefore C$ is a valid argument involving statement variables $p_1, p_2, \ldots, p_\ell\text{.}$ If we apply substitution $p_i\to B_i$ to each of $A_1, A_2, \ldots, A_m, C\text{,}$ for some collections of statements $B_1, B_2, \ldots, B_\ell\text{,}$ then the resulting argument is also valid.

Example $\PageIndex{1}$

Since modus tollens is a valid argument, using the substitution rule with the equivalences

$$\begin{equation*} r \land p \Leftrightarrow \neg (\neg r \lor \neg p) \Leftrightarrow \neg (r\rightarrow \neg p) \text{,} \end{equation*}$$

demonstrates that the following argument is also valid.

$\begin{aligned} &(p \leftrightarrow q) \rightarrow(r \rightarrow \neg p) \\ &r \wedge p \\ &\hline \neg(p \leftrightarrow q) \end{aligned}$