5.3: Substituting into an argument
Substituting into an argument does not change its validity.
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.
Since modus tollens is a valid argument, using the substitution rule with the equivalences
demonstrates that the following argument is also valid.
\(\begin{aligned}
&(p \leftrightarrow q) \rightarrow(r \rightarrow \neg p) \\
&r \wedge p \\
&\hline \neg(p \leftrightarrow q)
\end{aligned}\)