Prove the sequent is valid for the formulas 
Prove that if $\Gamma$ is a sequence of formulas, and the sequent $\Gamma, \phi \vdash \psi$ is valid then the sequent $\Gamma, \neg \psi \vdash \neg \phi$ is also valid. Do not use soundness/completeness.

I would like some assistance here. Without soundness, completeness how can I proceed further?
I see no way to proceed. Hints/ideas/theories appreciated!
 A: A sequent $\Gamma' \vdash \Delta$ is satisfied by an interprettaion $\mathcal I$ if either some formula in $\Gamma'$ is not satisfied by $\mathcal I$ or some formula in $\Delta$ is satisfied by $\mathcal I$.
A sequent is valid if it is satisfied by every interpretation.
Thus, we have to apply the above definition to the case when $\Gamma' = \Gamma \cup \{ \phi \}$ and $\Delta = \{ \psi \}$.
We assume that $Γ, \phi ⊢ \psi$ is valid and we can disregard the "context" $\Gamma$; for an interpretation $\mathcal I$ whatever, we have two cases to consider; either:
(i) $\phi$ is not satisfied by $\mathcal I$. 
In this case, we have that $\lnot \phi$ is satisfied and thus the sequent: $Γ, \lnot \psi ⊢ \lnot \phi$ is satisfied by $\mathcal I$.
Or:
(ii) $\psi$ is satisfied by $\mathcal I$. 
In this case, we have that $\lnot \psi$ is not satisfied and thus the sequent: $Γ, \lnot \psi ⊢ \lnot \phi$ is again satisfied by $\mathcal I$.
A: In natural deduction: See that $ \Gamma, \neg \psi, \phi \vdash \neg \psi$ (Reflexivity) and $ \Gamma, \neg \psi, \phi \vdash \psi$ (Monotonicity using the hypothesis), thus $ \Gamma, \neg \psi, \phi \vdash \bot$, using introduction of the implication, follows that $\Gamma, \neg \psi \vdash  \phi \to \bot$, but $\phi \to \bot = \neg \phi.$ To be clearer, you need especificate your rules.
