Theorem addcan2 ≪ | index | src | ≫

theorem addcan2 (a b c: nat): $ a + b = a + c <-> b = c $;

Step	Hyp	Ref	Expression
1		bitr	(a + b = a + c <-> b + a = c + a) -> (b + a = c + a <-> b = c) -> (a + b = a + c <-> b = c)
2		eqeq	a + b = b + a -> a + c = c + a -> (a + b = a + c <-> b + a = c + a)
3		addcom	a + b = b + a
4	2, 3	ax_mp	a + c = c + a -> (a + b = a + c <-> b + a = c + a)
5		addcom	a + c = c + a
6	4, 5	ax_mp	a + b = a + c <-> b + a = c + a
7	1, 6	ax_mp	(b + a = c + a <-> b = c) -> (a + b = a + c <-> b = c)
8		addcan1	b + a = c + a <-> b = c
9	7, 8	ax_mp	a + b = a + c <-> b = c

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_peano (peano2, peano5, addeq, add0, addS)