theorem eqthed (A: set) (G: wff) (a: nat) {x: nat}:
$ G -> (x e. A <-> x = a) $ >
$ G -> the A = a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
theid |
A == {z | z = a} -> the A = a |
| 2 |
|
eqeq1 |
z = x -> (z = a <-> x = a) |
| 3 |
2 |
elabe |
x e. {z | z = a} <-> x = a |
| 4 |
|
hyp h |
G -> (x e. A <-> x = a) |
| 5 |
3, 4 |
syl6bbr |
G -> (x e. A <-> x e. {z | z = a}) |
| 6 |
5 |
eqrd |
G -> A == {z | z = a} |
| 7 |
1, 6 |
syl |
G -> the A = 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)