Deduction of Modus Tollens

Question

I was wondering what is a deduction of Modus Tollens is.

However, there are only 3 axioms that I can use to proceed on the deduction.

The 3 axioms are in the link.

Answer 1

Axioms I and III look immediately usable.

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$$\begin{array}{ll}1.&\varphi&\text{as Hypothesis I}\\ 2.&(\lnot\psi\to\lnot\varphi)&\text{as Hypothesis II}\\ \hline 3.&(\lnot\psi\to\lnot\varphi)\to((\lnot\psi\to\varphi)\to\psi)&\text{by Axiom III}\\ 4.&(\lnot\psi\to\varphi)\to\psi&\text{using Modus Ponens (2, 3)}\\ 5.&\varphi\to(\lnot\psi\to\varphi)&\text{by Axiom I}\\ 6.\\7.\\8.\\9.\\ 10.&(\varphi\to\psi)&\text{using Modus Ponens (5, 9)}\\ 11.&\psi&\text{using Modus Ponens (1, 10)}\\ \Box\end{array}$$

$\Box$

You've just got to patch the hole in between, to show that: $$(\varphi\to(\lnot\psi\to\varphi))\to((\lnot\psi\to\varphi)\to\psi)\to(\varphi\to\psi))$$