Theorem truenat ≪ | index | src | ≫

theorem truenat (p: wff): $ true (nat p) <-> p $;

Step	Hyp	Ref	Expression
1		con1	(~p -> nat p = 0) -> ~nat p = 0 -> p
2	1	conv ne, true	(~p -> nat p = 0) -> true (nat p) -> p
3		ifneg	~p -> if p 1 0 = 0
4	3	conv nat	~p -> nat p = 0
5	2, 4	ax_mp	true (nat p) -> p
6		d1ne0	1 != 0
7		ifpos	p -> if p 1 0 = 1
8	7	conv nat	p -> nat p = 1
9	8	neeq1d	p -> (nat p != 0 <-> 1 != 0)
10	9	conv true	p -> (true (nat p) <-> 1 != 0)
11	6, 10	mpbiri	p -> true (nat p)
12	5, 11	ibii	true (nat p) <-> 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)