theorem eqrd2 (A B: set) (G: wff) {x y: nat}:
$ G -> (x, y e. A <-> x, y e. B) $ >
$ G -> A == B $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
axext2 |
A == B <-> A. x A. y (x, y e. A <-> x, y e. B) |
| 2 |
|
hyp h |
G -> (x, y e. A <-> x, y e. B) |
| 3 |
2 |
iald |
G -> A. y (x, y e. A <-> x, y e. B) |
| 4 |
3 |
iald |
G -> A. x A. y (x, y e. A <-> x, y e. B) |
| 5 |
1, 4 |
sylibr |
G -> A == B |
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
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)