theorem eqmsuc (a b n: nat): $ mod(n): suc a = suc b <-> mod(n): a = b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr3 |
(mod(n): a + 1 = b + 1 <-> mod(n): suc a = suc b) -> (mod(n): a + 1 = b + 1 <-> mod(n): a = b) -> (mod(n): suc a = suc b <-> mod(n): a = b) |
| 2 |
|
eqmeq |
n = n -> a + 1 = suc a -> b + 1 = suc b -> (mod(n): a + 1 = b + 1 <-> mod(n): suc a = suc b) |
| 3 |
|
eqid |
n = n |
| 4 |
2, 3 |
ax_mp |
a + 1 = suc a -> b + 1 = suc b -> (mod(n): a + 1 = b + 1 <-> mod(n): suc a = suc b) |
| 5 |
|
add12 |
a + 1 = suc a |
| 6 |
4, 5 |
ax_mp |
b + 1 = suc b -> (mod(n): a + 1 = b + 1 <-> mod(n): suc a = suc b) |
| 7 |
|
add12 |
b + 1 = suc b |
| 8 |
6, 7 |
ax_mp |
mod(n): a + 1 = b + 1 <-> mod(n): suc a = suc b |
| 9 |
1, 8 |
ax_mp |
(mod(n): a + 1 = b + 1 <-> mod(n): a = b) -> (mod(n): suc a = suc b <-> mod(n): a = b) |
| 10 |
|
eqmadd1 |
mod(n): a + 1 = b + 1 <-> mod(n): a = b |
| 11 |
9, 10 |
ax_mp |
mod(n): suc a = suc b <-> 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)