Theorem leaddsub ≪ | index | src | ≫

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

Step	Hyp	Ref	Expression
1		lenlt	a + b <= c <-> ~c < a + b
2		lenlt	a <= c - b <-> ~c - b < a
3		ltsubadd	b <= c -> (c - b < a <-> c < a + b)
4	3	bicomd	b <= c -> (c < a + b <-> c - b < a)
5	4	noteqd	b <= c -> (~c < a + b <-> ~c - b < a)
6	1, 2, 5	bitr4g	b <= c -> (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)