theorem elsbs {x: nat} (a: nat x) (b: nat) (A: set x):
$ b e. S[a / x] A <-> [a / x] b e. A $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(b e. S[a / x] A <-> [b / y] [a / x] y e. A) -> ([b / y] [a / x] y e. A <-> [a / x] b e. A) -> (b e. S[a / x] A <-> [a / x] b e. A) |
| 2 |
|
elab |
b e. {y | [a / x] y e. A} <-> [b / y] [a / x] y e. A |
| 3 |
2 |
conv sbs |
b e. S[a / x] A <-> [b / y] [a / x] y e. A |
| 4 |
1, 3 |
ax_mp |
([b / y] [a / x] y e. A <-> [a / x] b e. A) -> (b e. S[a / x] A <-> [a / x] b e. A) |
| 5 |
|
eleq1 |
y = b -> (y e. A <-> b e. A) |
| 6 |
5 |
sbeq2d |
y = b -> ([a / x] y e. A <-> [a / x] b e. A) |
| 7 |
6 |
sbe |
[b / y] [a / x] y e. A <-> [a / x] b e. A |
| 8 |
4, 7 |
ax_mp |
b e. S[a / x] A <-> [a / x] b e. 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)