Theorem boolnat ≪ | index | src | ≫

theorem boolnat (p: wff): $ bool (nat p) $;

Step	Hyp	Ref	Expression
1		d1lt2	1 < 2
2		ifpos	p -> if p 1 0 = 1
3	2	conv nat	p -> nat p = 1
4	3	lteq1d	p -> (nat p < 2 <-> 1 < 2)
5	4	conv bool	p -> (bool (nat p) <-> 1 < 2)
6	1, 5	mpbiri	p -> bool (nat p)
7		d0lt2	0 < 2
8		ifneg	~p -> if p 1 0 = 0
9	8	conv nat	~p -> nat p = 0
10	9	lteq1d	~p -> (nat p < 2 <-> 0 < 2)
11	10	conv bool	~p -> (bool (nat p) <-> 0 < 2)
12	7, 11	mpbiri	~p -> bool (nat p)
13	6, 12	cases	bool (nat 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)