theorem appslam (B: set) (a b: nat) {x: nat} (A: set x):
$ x = a -> A == B $ >
$ (\\ x, A) @ (a, b) = B @ b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
(\\ x, A) @ (a, b) = (S[a / x] A) @ b -> (S[a / x] A) @ b = B @ b -> (\\ x, A) @ (a, b) = B @ b |
| 2 |
|
appslams |
(\\ x, A) @ (a, b) = (S[a / x] A) @ b |
| 3 |
1, 2 |
ax_mp |
(S[a / x] A) @ b = B @ b -> (\\ x, A) @ (a, b) = B @ b |
| 4 |
|
appeq1 |
S[a / x] A == B -> (S[a / x] A) @ b = B @ b |
| 5 |
|
hyp h |
x = a -> A == B |
| 6 |
5 |
sbse |
S[a / x] A == B |
| 7 |
4, 6 |
ax_mp |
(S[a / x] A) @ b = B @ b |
| 8 |
3, 7 |
ax_mp |
(\\ x, A) @ (a, b) = B @ 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)