Theorem mulcan2 ≪ | index | src | ≫

theorem mulcan2 (a b c: nat): $ a != 0 -> (a * b = a * c <-> b = c) $;

Step	Hyp	Ref	Expression
1		eqeq	a * b = b * a -> a * c = c * a -> (a * b = a * c <-> b * a = c * a)
2		mulcom	a * b = b * a
3	1, 2	ax_mp	a * c = c * a -> (a * b = a * c <-> b * a = c * a)
4		mulcom	a * c = c * a
5	3, 4	ax_mp	a * b = a * c <-> b * a = c * a
6		mulcan1	a != 0 -> (b * a = c * a <-> b = c)
7	5, 6	syl5bb	a != 0 -> (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 (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)