theorem alim1 {x: nat} (a: wff) (b: wff x): $ A. x (a -> b) <-> a -> A. x b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
mpcom |
a -> (a -> b) -> b |
| 2 |
1 |
alimd |
a -> A. x (a -> b) -> A. x b |
| 3 |
2 |
com12 |
A. x (a -> b) -> a -> A. x b |
| 4 |
|
nfv |
F/ x a |
| 5 |
|
nfal1 |
F/ x A. x b |
| 6 |
4, 5 |
nfim |
F/ x a -> A. x b |
| 7 |
|
eal |
A. x b -> b |
| 8 |
7 |
imim2i |
(a -> A. x b) -> a -> b |
| 9 |
6, 8 |
ialdh |
(a -> A. x b) -> A. x (a -> b) |
| 10 |
3, 9 |
ibii |
A. x (a -> b) <-> a -> A. x 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_12)