I was trying to solve the following exercise.
For $\phi, \psi \in \mathcal{W}(P)$, $\phi$ and $\psi$ are logically equivalent if and only if $(\phi \leftrightarrow \psi)$ is a tautology. Let $\phi$ and $\psi$ be logically equivalent propositional formulas whose propositional variables are amongst $p_{1}, \ldots, p_{n} .$ Let $\eta_{1}, \ldots, \eta_{n}$ be propositional formulas and let $\theta$ and $\chi$ be the formulas obtained by replacing each occurrence of $p_{i}$ by $\eta_{i}$ in $\phi$ and $\psi$ respectively. Show that $\theta$ and $\chi$ are logically equivalent.
My attempt was to use induction on the construction of propositional formulas. That is, given a set $P$ of atoms and a 'property', the induction theorem that I was taught requires me to check:
The property holds for the atoms (including falsum)
Property holds for $\gamma \in \mathcal{W}(P)$ $\implies $ it holds for $\lnot \gamma \in \mathcal{W}(P)$
Property holds for $\gamma_1, \gamma_2 \in \mathcal{W}(P)$ $\implies $ it holds for $\gamma_1 \lor \gamma_2, \gamma_1 \land \gamma_2, \gamma_1 \rightarrow \gamma_2, \gamma_1 \leftrightarrow \gamma_2$
Then we can conclude that the property holds for all of $\mathcal{W}(P)$.
However, I was struggling to get the above problem into this framework. For example, what formula do I induct over? There is $\phi, \psi, \eta_i$, and I wasn't sure how to proceed. To be honest, after struggling for so long now I am having doubts that this is the correct approach for this problem. (obviously it need not be)
