Hilbert system equivalence

Question

Show that it’s provable in the Hilbert system that identity behaves as an equivalence relation, i.e., show that for any terms $t_1, t_2$, and $t_3$ the following hold.

(a) $\vdash_H t_1 \approx t_1$

(b) $\vdash_H t_1 \approx t_2 \rightarrow t_2 \approx t_1$

(c) $\vdash_H t_1 \approx t_2 \rightarrow t_2 \approx t_3 \rightarrow t_1 \approx t_3$

relevant axioms

1. $\alpha \rightarrow \big(\beta \rightarrow\alpha \big)$

2. $ \big({\alpha \rightarrow \big(\beta \rightarrow\gamma \big)}\big) \rightarrow \big({\big(\alpha \rightarrow\beta \big) \rightarrow \big(\alpha \rightarrow\gamma \big)} \big)$

3. $\big( \lnot \beta \rightarrow \lnot \alpha \big) \rightarrow \big( (\lnot \beta \rightarrow\alpha) \rightarrow \beta \big)$