Theorem lesubadd ≪ | index | src | ≫

theorem lesubadd (a b c: nat): $ a - b <= c <-> a <= c + b $;

Step	Hyp	Ref	Expression
1		bitr	(a - b <= c <-> a <= b + c) -> (a <= b + c <-> a <= c + b) -> (a - b <= c <-> a <= c + b)
2		lesubadd2	a - b <= c <-> a <= b + c
3	1, 2	ax_mp	(a <= b + c <-> a <= c + b) -> (a - b <= c <-> a <= c + b)
4		leeq2	b + c = c + b -> (a <= b + c <-> a <= c + b)
5		addcom	b + c = c + b
6	4, 5	ax_mp	a <= b + c <-> a <= c + b
7	3, 6	ax_mp	a - b <= c <-> a <= c + b

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, the0), axs_peano (peano1, peano2, peano5, addeq, add0, addS)