theorem elima (A F: set) (b: nat) {x: nat}:
$ b e. F '' A <-> E. x (x e. A /\ x, b e. F) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
preq2 |
y = b -> x, y = x, b |
| 2 |
1 |
eleq1d |
y = b -> (x, y e. F <-> x, b e. F) |
| 3 |
2 |
aneq2d |
y = b -> (x e. A /\ x, y e. F <-> x e. A /\ x, b e. F) |
| 4 |
3 |
exeqd |
y = b -> (E. x (x e. A /\ x, y e. F) <-> E. x (x e. A /\ x, b e. F)) |
| 5 |
4 |
elabe |
b e. {y | E. x (x e. A /\ x, y e. F)} <-> E. x (x e. A /\ x, b e. F) |
| 6 |
5 |
conv Im |
b e. F '' A <-> E. x (x e. A /\ x, b e. F) |
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)