Theorem subadd ≪ | index | src | ≫

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

Step	Hyp	Ref	Expression
1		eqcom	a - b = c -> c = a - b
2		addeq2	c = a - b -> b + c = b + (a - b)
3	2	eqeq1d	c = a - b -> (b + c = a <-> b + (a - b) = a)
4	1, 3	rsyl	a - b = c -> (b + c = a <-> b + (a - b) = a)
5		pncan3	b <= a -> b + (a - b) = a
6	4, 5	syl5ibrcom	b <= a -> a - b = c -> b + c = a
7		eqsub2	b + c = a -> a - b = c
8	7	a1i	b <= a -> b + c = a -> a - b = c
9	6, 8	ibid	b <= a -> (a - b = c <-> b + c = a)

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)