theorem appslams (a b: nat) {x: nat} (A: set x):
$ (\\ x, A) @ (a, b) = (S[a / x] A) @ b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
theeq |
{a1 | (a, b), a1 e. (\\ x, A)} == {a1 | b, a1 e. S[a / x] A} -> the {a1 | (a, b), a1 e. (\\ x, A)} = the {a1 | b, a1 e. S[a / x] A} |
| 2 |
1 |
conv app |
{a1 | (a, b), a1 e. (\\ x, A)} == {a1 | b, a1 e. S[a / x] A} -> (\\ x, A) @ (a, b) = (S[a / x] A) @ b |
| 3 |
|
prelslams |
(a, b), a1 e. (\\ x, A) <-> b, a1 e. S[a / x] A |
| 4 |
3 |
abeqi |
{a1 | (a, b), a1 e. (\\ x, A)} == {a1 | b, a1 e. S[a / x] A} |
| 5 |
2, 4 |
ax_mp |
(\\ x, A) @ (a, b) = (S[a / x] 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)