Theorem zaddcan1 ≪ | index | src | ≫

theorem zaddcan1 (a b c: nat): $ a +Z c = b +Z c <-> a = b $;

Step	Hyp	Ref	Expression
1		zpncan	a +Z c -Z c = a
2		zpncan	b +Z c -Z c = b
3		zsubeq1	a +Z c = b +Z c -> a +Z c -Z c = b +Z c -Z c
4	2, 3	syl6eq	a +Z c = b +Z c -> a +Z c -Z c = b
5	1, 4	syl5eqr	a +Z c = b +Z c -> a = b
6		zaddeq1	a = b -> a +Z c = b +Z c
7	5, 6	ibii	a +Z c = b +Z c <-> a = 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, muleq, add0, addS, mul0, mulS)