Proof using Modus Ponens, abduction and contradiction

Question

We have

$Κ_1: φ→χ$

$Κ_2: ¬ψ→φ$

$Κ_3: ¬χ$

$Κ_4: ψ$

And we must prove that $[Κ_1,Κ_2,Κ_3] ⊢ K_4$.

Can i use an abduction like this?

$[Κ_1,Κ_2,Κ_3] ⊢ ¬ψ→¬ψ ⇒ [Κ_1,Κ_2,Κ_3,¬ψ] ⊢ ¬ψ$

1. φ→χ Hypothesis

2. ¬ψ→φ Hypothesis

3. ¬χ Hypothesis

4. ¬ψ Hypothesis

5. φ Modus Ponens from 2, 4

6. χ Modus Ponens from 1, 5

7. Contradiction (3, 6)

Answer 1

What your derivation shows is that:

$[Κ_1,Κ_2,Κ_3,\neg \psi] \vdash \bot$

and that means that:

$[Κ_1,Κ_2,Κ_3] \vdash \psi$

i.e.

$[Κ_1,Κ_2,Κ_3] \vdash K_4$