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