Theorem nateq1 ≪ | index | src | ≫

theorem nateq1 (p: wff): $ nat p = 1 <-> p $;

Step	Hyp	Ref	Expression
1		bitr3	(true (nat p) <-> nat p = 1) -> (true (nat p) <-> p) -> (nat p = 1 <-> p)
2		dftrue2	bool (nat p) -> (true (nat p) <-> nat p = 1)
3		boolnat	bool (nat p)
4	2, 3	ax_mp	true (nat p) <-> nat p = 1
5	1, 4	ax_mp	(true (nat p) <-> p) -> (nat p = 1 <-> p)
6		truenat	true (nat p) <-> p
7	5, 6	ax_mp	nat p = 1 <-> 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, peano2, peano5, addeq, add0, addS)