theorem cbvxabs (A: set) {x y: nat} (B: set x):
$ X\ x e. A, B == X\ y e. A, (S[y / x] B) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eleq2 |
S[fst z / x] B == S[fst z / y] S[y / x] B -> (snd z e. S[fst z / x] B <-> snd z e. S[fst z / y] S[y / x] B) |
| 2 |
|
eqscom |
S[fst z / y] S[y / x] B == S[fst z / x] B -> S[fst z / x] B == S[fst z / y] S[y / x] B |
| 3 |
|
sbsco |
S[fst z / y] S[y / x] B == S[fst z / x] B |
| 4 |
2, 3 |
ax_mp |
S[fst z / x] B == S[fst z / y] S[y / x] B |
| 5 |
1, 4 |
ax_mp |
snd z e. S[fst z / x] B <-> snd z e. S[fst z / y] S[y / x] B |
| 6 |
5 |
aneq2i |
fst z e. A /\ snd z e. S[fst z / x] B <-> fst z e. A /\ snd z e. S[fst z / y] S[y / x] B |
| 7 |
6 |
abeqi |
{z | fst z e. A /\ snd z e. S[fst z / x] B} == {z | fst z e. A /\ snd z e. S[fst z / y] S[y / x] B} |
| 8 |
7 |
conv xab |
X\ x e. A, B == X\ y e. A, (S[y / x] 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)