Theorem zaddcan2 ≪ | index | src | ≫ 

theorem zaddcan2 (a b c: nat): $ a +Z b = a +Z c <-> b = c $;
StepHypRefExpression
1 bitr 
(a +Z b = a +Z c <-> b +Z a = c +Z a) -> (b +Z a = c +Z a <-> b = c) -> (a +Z b = a +Z c <-> b = c)
2 eqeq 
a +Z b = b +Z a -> a +Z c = c +Z a -> (a +Z b = a +Z c <-> b +Z a = c +Z a)
3 zaddcom 
a +Z b = b +Z a
4 2, 3 ax_mp 
a +Z c = c +Z a -> (a +Z b = a +Z c <-> b +Z a = c +Z a)
5 zaddcom 
a +Z c = c +Z a
6 4, 5 ax_mp 
a +Z b = a +Z c <-> b +Z a = c +Z a
7 1, 6 ax_mp 
(b +Z a = c +Z a <-> b = c) -> (a +Z b = a +Z c <-> b = c)
8 zaddcan1 
b +Z a = c +Z a <-> b = c
9 7, 8 ax_mp 
a +Z b = a +Z 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_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)