theorem elimai (A F: set) (a b: nat): $ a, b e. F -> a e. A -> b e. F '' A $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
elima |
b e. F '' A <-> E. x (x e. A /\ x, b e. F) |
| 2 |
|
eleq1 |
x = a -> (x e. A <-> a e. A) |
| 3 |
|
preq1 |
x = a -> x, b = a, b |
| 4 |
3 |
eleq1d |
x = a -> (x, b e. F <-> a, b e. F) |
| 5 |
2, 4 |
aneqd |
x = a -> (x e. A /\ x, b e. F <-> a e. A /\ a, b e. F) |
| 6 |
5 |
iexe |
a e. A /\ a, b e. F -> E. x (x e. A /\ x, b e. F) |
| 7 |
1, 6 |
sylibr |
a e. A /\ a, b e. F -> b e. F '' A |
| 8 |
7 |
expcom |
a, b e. F -> a e. A -> b e. F '' A |
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
(peano2,
addeq,
muleq)