I have a question on the proof for uniform substitution preserving validity, in the basic modal system $\mathbf{K}$.
We are assuming that for an arbitrary Kripke model $\mathcal{M}$ and world $s$ we have that $\mathcal{M},s\Vdash\varphi$, where $\varphi$ is any modal formula in the language, and want to prove that $\mathcal{M},s\Vdash\varphi[p/\psi]$ where $\varphi[p/\psi]$ is the formula resulting from substituting every instance of $p$ in $\varphi$ for $\psi$. The proof is now by induction on the structure of $\varphi$. My question is for the implication case:
Suppose $\varphi$ is $\zeta_1\rightarrow\zeta_2$. We have that $\mathcal{M},s\Vdash\zeta_1\rightarrow\zeta_2$, i.e., $\mathcal{M},s\not\Vdash\zeta_1$ or $\mathcal{M},s\Vdash\zeta_2$.
- If $\mathcal{M},s\Vdash\zeta_2$, by induction hypothesis $\mathcal{M},s\Vdash\zeta_2[p/\psi]$, hence $\mathcal{M},s\Vdash\zeta_1[p/\psi]\rightarrow\zeta_2[p/\psi]$.
- If $\mathcal{M},s\not\Vdash\zeta_1$ can we say that by induction hypothesis $\mathcal{M},s\not\Vdash\zeta_1[p/\psi]$? It doesn't make much sense to me since the I.H. seems to be an implication. How do we go on from here?