Theorem addsub ≪ | index | src | ≫

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

Step	Hyp	Ref	Expression
1		pncan	a - c + b + c - c = a - c + b
2		addrass	a - c + c + b = a - c + b + c
3		npcan	c <= a -> a - c + c = a
4	3	eqcomd	c <= a -> a = a - c + c
5	4	addeq1d	c <= a -> a + b = a - c + c + b
6	2, 5	syl6eq	c <= a -> a + b = a - c + b + c
7	6	subeq1d	c <= a -> a + b - c = a - c + b + c - c
8	1, 7	syl6eq	c <= a -> 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 (peano2, peano5, addeq, add0, addS)