theorem rexpim1 {x y: nat} (a: wff x) (b: wff y) (c: wff x y):
$ E. x (a /\ (P. y b -> c)) -> (P. y b -> E. x (a /\ c)) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
anrl |
a /\ (E. y b /\ A. y (b -> c)) -> E. y b |
| 2 |
1 |
conv pim |
a /\ (P. y b -> c) -> E. y b |
| 3 |
2 |
eex |
E. x (a /\ (P. y b -> c)) -> E. y b |
| 4 |
|
anim2 |
((P. y b -> c) -> A. y (b -> c)) -> a /\ (P. y b -> c) -> a /\ A. y (b -> c) |
| 5 |
|
anr |
E. y b /\ A. y (b -> c) -> A. y (b -> c) |
| 6 |
5 |
conv pim |
(P. y b -> c) -> A. y (b -> c) |
| 7 |
4, 6 |
ax_mp |
a /\ (P. y b -> c) -> a /\ A. y (b -> c) |
| 8 |
7 |
eximi |
E. x (a /\ (P. y b -> c)) -> E. x (a /\ A. y (b -> c)) |
| 9 |
|
rexral |
E. x (a /\ A. y (b -> c)) -> A. y (b -> E. x (a /\ c)) |
| 10 |
8, 9 |
rsyl |
E. x (a /\ (P. y b -> c)) -> A. y (b -> E. x (a /\ c)) |
| 11 |
3, 10 |
iand |
E. x (a /\ (P. y b -> c)) -> E. y b /\ A. y (b -> E. x (a /\ c)) |
| 12 |
11 |
conv pim |
E. x (a /\ (P. y b -> c)) -> (P. y b -> E. x (a /\ c)) |
Axiom use
axs_prop_calc
(ax_1,
ax_2,
ax_3,
ax_mp),
axs_pred_calc
(ax_gen,
ax_4,
ax_5,
ax_6,
ax_7,
ax_10,
ax_11,
ax_12)