theorem poweqd (_G: wff) (_a1 _a2 _b1 _b2: nat):
$ _G -> _a1 = _a2 $ >
$ _G -> _b1 = _b2 $ >
$ _G -> _a1 ^ _b1 = _a2 ^ _b2 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqidd |
_G -> 1 = 1 |
| 2 |
|
hyp _ah |
_G -> _a1 = _a2 |
| 3 |
|
eqidd |
_G -> n = n |
| 4 |
2, 3 |
muleqd |
_G -> _a1 * n = _a2 * n |
| 5 |
4 |
lameqd |
_G -> \ n, _a1 * n == \ n, _a2 * n |
| 6 |
|
hyp _bh |
_G -> _b1 = _b2 |
| 7 |
1, 5, 6 |
receqd |
_G -> rec 1 (\ n, _a1 * n) _b1 = rec 1 (\ n, _a2 * n) _b2 |
| 8 |
7 |
conv pow |
_G -> _a1 ^ _b1 = _a2 ^ _b2 |
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
(peano2,
addeq,
muleq)