Theorem nateq0 ≪ | index | src | ≫

theorem nateq0 (p: wff): $ nat p = 0 <-> ~p $;

Step	Hyp	Ref	Expression
1		con2b	(p <-> ~nat p = 0) -> (nat p = 0 <-> ~p)
2		bicom	(~nat p = 0 <-> p) -> (p <-> ~nat p = 0)
3		truenat	true (nat p) <-> p
4	3	conv ne, true	~nat p = 0 <-> p
5	2, 4	ax_mp	p <-> ~nat p = 0
6	1, 5	ax_mp	nat p = 0 <-> ~p

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8), axs_the (theid), axs_peano (peano1)