theorem elwrite (F: set) (a b x y: nat):
$ x, y e. write F a b <-> ifp (x = a) (y = b) (x, y e. F) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
anl |
a1 = x /\ a2 = y -> a1 = x |
| 2 |
1 |
eqeq1d |
a1 = x /\ a2 = y -> (a1 = a <-> x = a) |
| 3 |
|
anr |
a1 = x /\ a2 = y -> a2 = y |
| 4 |
3 |
eqeq1d |
a1 = x /\ a2 = y -> (a2 = b <-> y = b) |
| 5 |
|
preq |
a1 = x -> a2 = y -> a1, a2 = x, y |
| 6 |
5 |
imp |
a1 = x /\ a2 = y -> a1, a2 = x, y |
| 7 |
6 |
eleq1d |
a1 = x /\ a2 = y -> (a1, a2 e. F <-> x, y e. F) |
| 8 |
2, 4, 7 |
ifpeqd |
a1 = x /\ a2 = y -> (ifp (a1 = a) (a2 = b) (a1, a2 e. F) <-> ifp (x = a) (y = b) (x, y e. F)) |
| 9 |
8 |
elabed |
a1 = x -> (y e. {a2 | ifp (a1 = a) (a2 = b) (a1, a2 e. F)} <-> ifp (x = a) (y = b) (x, y e. F)) |
| 10 |
9 |
elsabe |
x, y e. S\ a1, {a2 | ifp (a1 = a) (a2 = b) (a1, a2 e. F)} <-> ifp (x = a) (y = b) (x, y e. F) |
| 11 |
10 |
conv write |
x, y e. write F a b <-> ifp (x = a) (y = b) (x, y e. F) |
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)