theorem addrass (a b c: nat): $ a + b + c = a + c + b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
a + b + c = a + (b + c) -> a + (b + c) = a + c + b -> a + b + c = a + c + b |
| 2 |
|
addass |
a + b + c = a + (b + c) |
| 3 |
1, 2 |
ax_mp |
a + (b + c) = a + c + b -> a + b + c = a + c + b |
| 4 |
|
eqtr4 |
a + (b + c) = a + (c + b) -> a + c + b = a + (c + b) -> a + (b + c) = a + c + b |
| 5 |
|
addeq2 |
b + c = c + b -> a + (b + c) = a + (c + b) |
| 6 |
|
addcom |
b + c = c + b |
| 7 |
5, 6 |
ax_mp |
a + (b + c) = a + (c + b) |
| 8 |
4, 7 |
ax_mp |
a + c + b = a + (c + b) -> a + (b + c) = a + c + b |
| 9 |
|
addass |
a + c + b = a + (c + b) |
| 10 |
8, 9 |
ax_mp |
a + (b + c) = a + c + b |
| 11 |
3, 10 |
ax_mp |
a + b + c = a + c + 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_peano
(peano2,
peano5,
addeq,
add0,
addS)