theorem add22 (a: nat): $ a + 2 = suc (suc a) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
a + 2 = suc (a + 1) -> suc (a + 1) = suc (suc a) -> a + 2 = suc (suc a) |
| 2 |
|
addS |
a + suc 1 = suc (a + 1) |
| 3 |
2 |
conv d2 |
a + 2 = suc (a + 1) |
| 4 |
1, 3 |
ax_mp |
suc (a + 1) = suc (suc a) -> a + 2 = suc (suc a) |
| 5 |
|
suceq |
a + 1 = suc a -> suc (a + 1) = suc (suc a) |
| 6 |
|
add12 |
a + 1 = suc a |
| 7 |
5, 6 |
ax_mp |
suc (a + 1) = suc (suc a) |
| 8 |
4, 7 |
ax_mp |
a + 2 = suc (suc a) |
Axiom use
axs_prop_calc
(ax_1,
ax_2,
ax_3,
ax_mp),
axs_pred_calc
(ax_gen,
ax_4,
ax_5,
ax_6,
ax_7),
axs_peano
(peano2,
add0,
addS)