theorem eqsub2 (a b c: nat): $ a + b = c -> c - a = b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
addeq2 |
y = x -> a + y = a + x |
| 2 |
1 |
eqeq1d |
y = x -> (a + y = c <-> a + x = c) |
| 3 |
2 |
elabe |
x e. {y | a + y = c} <-> a + x = c |
| 4 |
|
addcan2 |
a + x = a + b <-> x = b |
| 5 |
|
eqeq2 |
a + b = c -> (a + x = a + b <-> a + x = c) |
| 6 |
5 |
bicomd |
a + b = c -> (a + x = c <-> a + x = a + b) |
| 7 |
4, 6 |
syl6bb |
a + b = c -> (a + x = c <-> x = b) |
| 8 |
3, 7 |
syl5bb |
a + b = c -> (x e. {y | a + y = c} <-> x = b) |
| 9 |
8 |
eqthed |
a + b = c -> the {y | a + y = c} = b |
| 10 |
9 |
conv sub |
a + b = c -> c - 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),
axs_peano
(peano2,
peano5,
addeq,
add0,
addS)