Prove that if $ \Gamma $ is inconsistent, then $\Gamma \vdash \beta $ for every formula $\beta$ Considering that $\Gamma$ is inconsistent if $ \Gamma \vdash ¬(\alpha \rightarrow \alpha) $ for some formula $\alpha$.

How to prove that if $ \Gamma $ is inconsistent, then $ \Gamma \vdash \beta $?

Do I need to assume that a proof of  $ ¬(\alpha \rightarrow \alpha) $ exists and try to find $\beta$ using axioms? Is it possible?

List of axioms:

Axioms
If you can prove $\Gamma \vdash ¬(\alpha \rightarrow \alpha)$, you can say $$\Gamma,\lnot \beta \vdash ¬(\alpha \rightarrow \alpha) \\ \Gamma \vdash (\lnot \beta \implies¬(\alpha \rightarrow \alpha)) \\ \Gamma \vdash \beta$$
Hint: Prove that, in Polish notation, $\vdash$CpCNpq.

More fully:

axiom 1 CpCqp
axiom 2 CCpCqrCCpqCpr
axiom 3 CCpqCCpNqNp
axiom 4 CNNpp
2 r/p 5 CCpCqpCCpqCpp
5, 1  6 CCpqCpp
6 q/Cqp 7 CCpCqpCpp
7, 1    8 Cpp
hypothesis  9 | p
hypothesis  10 || Np
1 q/Nq      11 || CpCNqp
1 q/Nq p/Np 12 || CNpCNqNp
11, 9       13 || CNqp
12, 10      14 || CNqNp
3 p/Nq, q/p 15 || CCNqpCCNqNpNNq
15, 14      16 || CCNqNpNNq
16, 14      17 || NNq
4 p/q       18 || CNNqq
18, 17      19 || q
10-19       20 | CNpq
9-20        21 CpCNpq
21 p/Cpp    22 CCppCNCppq
22, 8       23 CNCppq


Thus, given that $\Gamma$ $\vdash$ NCpp for some formula p, since 23 holds, it follows that $\Gamma$ $\vdash$ q.