theorem rlamfunc2 (A B: set) (a: nat) {x: nat} (b: nat x):
$ A == a -> (func (\. x e. a, b) A B <-> A. x (x e. A -> b e. B)) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
rlamfunc |
func (\. x e. a, b) a B <-> A. x (x e. a -> b e. B) |
| 2 |
|
id |
A == a -> A == a |
| 3 |
2 |
funceq2d |
A == a -> (func (\. x e. a, b) A B <-> func (\. x e. a, b) a B) |
| 4 |
2 |
eleq2d |
A == a -> (x e. A <-> x e. a) |
| 5 |
4 |
imeq1d |
A == a -> (x e. A -> b e. B <-> x e. a -> b e. B) |
| 6 |
5 |
aleqd |
A == a -> (A. x (x e. A -> b e. B) <-> A. x (x e. a -> b e. B)) |
| 7 |
3, 6 |
bieqd |
A == a -> (func (\. x e. a, b) A B <-> A. x (x e. A -> b e. B) <-> (func (\. x e. a, b) a B <-> A. x (x e. a -> b e. B))) |
| 8 |
1, 7 |
mpbiri |
A == a -> (func (\. x e. a, b) A B <-> A. x (x e. A -> b e. 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)