As the title says: I want to prove by induction that $\Gamma \vdash \varphi \Rightarrow \Gamma[x/c] \vdash \varphi[x/c]$.
I’m struggling with how to write this proof. I think I need to do induction on the size of the derivation. The base case would be derivations of size 1, ie. $size(\mathcal{D}) = 1$. This, I imagine, would include $\vdash \varphi$ for tautological $\varphi$ and $\sigma \vdash \sigma$. This might not be right, but that’s not my biggest problem. I can’t figure out how to formulate an inductive hypothesis. Any help appreciated!