Theorem eqzsub ≪ | index | src | ≫ 

theorem eqzsub (a b c: nat): $ a -Z c = b <-> b +Z c = a $;
StepHypRefExpression
1 bitr 
(a -Z c = b <-> a = b +Z c) -> (a = b +Z c <-> b +Z c = a) -> (a -Z c = b <-> b +Z c = a)
2 znpcan 
a -Z c +Z c = a
3 zaddeq1 
a -Z c = b -> a -Z c +Z c = b +Z c
4 2, 3 syl5eqr 
a -Z c = b -> a = b +Z c
5 zpncan 
b +Z c -Z c = b
6 zsubeq1 
a = b +Z c -> a -Z c = b +Z c -Z c
7 5, 6 syl6eq 
a = b +Z c -> a -Z c = b
8 4, 7 ibii 
a -Z c = b <-> a = b +Z c
9 1, 8 ax_mp 
(a = b +Z c <-> b +Z c = a) -> (a -Z c = b <-> b +Z c = a)
10 eqcomb 
a = b +Z c <-> b +Z c = a
11 9, 10 ax_mp 
a -Z c = b <-> b +Z 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 (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)