Let $\Phi \subseteq L_{\Sigma}$ where $\Sigma$ is the set of all propositional symbols $p, q, r,\dots$ and $L_{\Sigma}$ de set of well formed formulas formed from $\Sigma$. Show that $\Phi \cup \{ \varphi_1,\dots, \varphi_n\} \models \psi$ implies $\Phi \models \varphi_1 \rightarrow \dots \rightarrow \varphi_n \rightarrow \psi$
I tried to prove it using induction on $n$ but I got a little confused. Here is what I did:
Take a valuation $v$ such that $v \models \Phi$. What I want to see is that $v \models \varphi_1 \rightarrow \dots \rightarrow \varphi_n \rightarrow \psi$ also holds. If $n = 1$, then we have only two cases:
In the first one, $v \not \models \varphi_1$, so $v \models \varphi_1 \rightarrow \psi$ and we are done. The other possible case is where $v \models \varphi_1$. But then $v \models \Phi$ and $v \models \varphi_1$, so $v \models \Phi \cup \{ \varphi_1 \}$, and by hypothesis, then $v \models \psi$, so $v \models \varphi$ and $v \models \psi$, therefore, $v \models \varphi_1 \rightarrow \psi$.
Now this is where I got stuck. I don't know how to proceed from there. I tried to do the same thing distinguishing cases but I think it is wrong. Like before, there can be two cases: $v \models \varphi_{n+1}$ or $v \not \models \varphi_{n+1}$.
If the case is the latter, then $v \models \varphi_{n+1} \rightarrow \psi$, but can I use the induction hypothesis to conclude that $$v \models \varphi_1 \rightarrow \dots \rightarrow \varphi_n \rightarrow \varphi_{n+1} \rightarrow \psi?$$ And again, if $v \models \varphi_{n+1}$ and having that $v \models \varphi_1 \rightarrow \dots \rightarrow \varphi_n \rightarrow \psi$, can I conclude that $$v \models \varphi_1\rightarrow \dots \rightarrow \varphi_n \rightarrow \varphi_{n+1} \rightarrow\psi?$$ Thank you very much for your help!
Minor edit I'm assuming that the conditional $\rightarrow$ associates to the right, that is, $\varphi_1 \rightarrow \varphi_2 \rightarrow \varphi_3 \equiv \varphi_1 \rightarrow\left( \varphi_2 \rightarrow \varphi_3 \right)$ and so on.