# Using $(\neg \psi \to \neg \phi )\to (\phi \to \psi )$ as axiom3 of $L$, show $\vdash (((\psi \to \phi )\rightarrow \psi )\rightarrow \psi )$？

How to show $$\vdash (((\psi \to \phi )\rightarrow \psi )\rightarrow \psi )$$？Equivalently,$$((\psi \to \phi )\rightarrow \psi )\vdash \psi$$?Of course, by formal language $$L$$. The Axiom3 of $$L$$ is $$(\neg \psi \to \neg \phi )\to (\phi \to \psi )$$--not same to the answer already put on website. This question came from Logic for mathematicians written by A. G. Hammilton, in the chapter 2 exercise3.c.

No big change needed...

In A.G. Hammilton, Logic for mathematicians's proof system we have:

Example 2.7 (page 31): $$\vdash (\phi\to \phi)$$.

Prop.2.8 (Deduction Theorem, page 32).

Prop.2.11 (page 35): (a) $$\vdash (\lnot \phi \to (\phi \to \psi))$$ and (b) $$\vdash ((\lnot \phi\to \phi) \to \phi)$$.

Now for the main proof :

1. $$(ϕ → ψ) → ϕ$$ --- premise

2. $$\lnot \phi$$ --- premise

3. $$\vdash \lnot \phi \to (\phi \to \psi)$$ --- Prop.2.11 (a)

4. $$\phi \to \psi$$ --- from 2) and 3) by MP

5. $$\phi$$ --- from 1) and 4) by mp

6. $$(ϕ → ψ) → ϕ \vdash \lnot \phi \to \phi$$ --- from 2) and 5) by Deduction Th

7. $$(ϕ → ψ) → ϕ \vdash ϕ$$ --- Prop.2.11 (b) and MP

$$\vdash ((ϕ → ψ ) → ϕ) → ϕ$$ --- from t) by Deduction Th.

• Oh, I see. Thanks! Commented Nov 9, 2020 at 10:30
• @yinfeng - If you are satisfied with the answer, you can accept it. Commented Nov 9, 2020 at 10:34