theorem b0can (a b: nat): $ b0 a = b0 b <-> a = b $;
Step | Hyp | Ref | Expression |
1 |
|
eqeq |
b0 a // 2 = a -> b0 b // 2 = b -> (b0 a // 2 = b0 b // 2 <-> a = b) |
2 |
|
b0div2 |
b0 a // 2 = a |
3 |
1, 2 |
ax_mp |
b0 b // 2 = b -> (b0 a // 2 = b0 b // 2 <-> a = b) |
4 |
|
b0div2 |
b0 b // 2 = b |
5 |
3, 4 |
ax_mp |
b0 a // 2 = b0 b // 2 <-> a = b |
6 |
|
diveq1 |
b0 a = b0 b -> b0 a // 2 = b0 b // 2 |
7 |
5, 6 |
sylib |
b0 a = b0 b -> a = b |
8 |
|
b0eq |
a = b -> b0 a = b0 b |
9 |
7, 8 |
ibii |
b0 a = b0 b <-> 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)