pub theorem writeNe (F: set) (a b x: nat):
$ x != a -> write F a b @ x = F @ x $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqapp |
A. y (x, y e. write F a b <-> x, y e. F) -> write F a b @ x = F @ x |
| 2 |
|
elwrite |
x, y e. write F a b <-> ifp (x = a) (y = b) (x, y e. F) |
| 3 |
|
ifpneg |
~x = a -> (ifp (x = a) (y = b) (x, y e. F) <-> x, y e. F) |
| 4 |
3 |
conv ne |
x != a -> (ifp (x = a) (y = b) (x, y e. F) <-> x, y e. F) |
| 5 |
2, 4 |
syl5bb |
x != a -> (x, y e. write F a b <-> x, y e. F) |
| 6 |
5 |
iald |
x != a -> A. y (x, y e. write F a b <-> x, y e. F) |
| 7 |
1, 6 |
syl |
x != a -> write F a b @ x = F @ x |
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)