theorem ralan {x: nat} (a b c: wff x):
$ A. x (a -> b /\ c) <-> A. x (a -> b) /\ A. x (a -> c) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(A. x (a -> b /\ c) <-> A. x ((a -> b) /\ (a -> c))) ->
(A. x ((a -> b) /\ (a -> c)) <-> A. x (a -> b) /\ A. x (a -> c)) ->
(A. x (a -> b /\ c) <-> A. x (a -> b) /\ A. x (a -> c)) |
| 2 |
|
imandi |
a -> b /\ c <-> (a -> b) /\ (a -> c) |
| 3 |
2 |
aleqi |
A. x (a -> b /\ c) <-> A. x ((a -> b) /\ (a -> c)) |
| 4 |
1, 3 |
ax_mp |
(A. x ((a -> b) /\ (a -> c)) <-> A. x (a -> b) /\ A. x (a -> c)) -> (A. x (a -> b /\ c) <-> A. x (a -> b) /\ A. x (a -> c)) |
| 5 |
|
alan |
A. x ((a -> b) /\ (a -> c)) <-> A. x (a -> b) /\ A. x (a -> c) |
| 6 |
4, 5 |
ax_mp |
A. x (a -> b /\ c) <-> A. x (a -> b) /\ A. x (a -> c) |
Axiom use
axs_prop_calc
(ax_1,
ax_2,
ax_3,
ax_mp),
axs_pred_calc
(ax_gen,
ax_4)