Theorem eqmadd1 ≪ | index | src | ≫ 

theorem eqmadd1 (a b c n: nat): $ mod(n): a + c = b + c <-> mod(n): a = b $;
StepHypRefExpression
1 eor 
(a <= b -> (mod(n): a + c = b + c <-> mod(n): a = b)) ->
  (b <= a -> (mod(n): a + c = b + c <-> mod(n): a = b)) ->
  a <= b \/ b <= a ->
  (mod(n): a + c = b + c <-> mod(n): a = b)
2 eqmaddlem 
a <= b -> (mod(n): a + c = b + c <-> mod(n): a = b)
3 1, 2 ax_mp 
(b <= a -> (mod(n): a + c = b + c <-> mod(n): a = b)) -> a <= b \/ b <= a -> (mod(n): a + c = b + c <-> mod(n): a = b)
4 eqmcomb 
mod(n): a + c = b + c <-> mod(n): b + c = a + c
5 eqmcomb 
mod(n): b = a <-> mod(n): a = b
6 eqmaddlem 
b <= a -> (mod(n): b + c = a + c <-> mod(n): b = a)
7 5, 6 syl6bb 
b <= a -> (mod(n): b + c = a + c <-> mod(n): a = b)
8 4, 7 syl5bb 
b <= a -> (mod(n): a + c = b + c <-> mod(n): a = b)
9 3, 8 ax_mp 
a <= b \/ b <= a -> (mod(n): a + c = b + c <-> mod(n): a = b)
10 leorle 
a <= b \/ b <= a
11 9, 10 ax_mp 
mod(n): a + c = b + c <-> mod(n): 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)