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theorem eqid (a: nat): $ a = a $;

Step	Hyp	Ref	Expression
1		imidm	(x = a -> x = a -> a = a) -> x = a -> a = a
2		ax_7	x = a -> x = a -> a = a
3	1, 2	ax_mp	x = a -> a = a
4	3	eex	E. x x = a -> a = a
5		ax_6	E. x x = a
6	4, 5	ax_mp	a = a

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7)