theorem grecaux1eq (_K1 _K2: set) (_x1 _x2 _z1 _z2 _n1 _n2: nat):
$ _K1 == _K2 ->
_x1 = _x2 ->
_z1 = _z2 ->
_n1 = _n2 ->
grecaux1 _K1 _x1 _z1 _n1 = grecaux1 _K2 _x2 _z2 _n2 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
anl |
_K1 == _K2 /\ _x1 = _x2 -> _K1 == _K2 |
| 2 |
1 |
anwl |
_K1 == _K2 /\ _x1 = _x2 /\ _z1 = _z2 -> _K1 == _K2 |
| 3 |
2 |
anwl |
_K1 == _K2 /\ _x1 = _x2 /\ _z1 = _z2 /\ _n1 = _n2 -> _K1 == _K2 |
| 4 |
|
anr |
_K1 == _K2 /\ _x1 = _x2 -> _x1 = _x2 |
| 5 |
4 |
anwl |
_K1 == _K2 /\ _x1 = _x2 /\ _z1 = _z2 -> _x1 = _x2 |
| 6 |
5 |
anwl |
_K1 == _K2 /\ _x1 = _x2 /\ _z1 = _z2 /\ _n1 = _n2 -> _x1 = _x2 |
| 7 |
|
anr |
_K1 == _K2 /\ _x1 = _x2 /\ _z1 = _z2 -> _z1 = _z2 |
| 8 |
7 |
anwl |
_K1 == _K2 /\ _x1 = _x2 /\ _z1 = _z2 /\ _n1 = _n2 -> _z1 = _z2 |
| 9 |
|
anr |
_K1 == _K2 /\ _x1 = _x2 /\ _z1 = _z2 /\ _n1 = _n2 -> _n1 = _n2 |
| 10 |
3, 6, 8, 9 |
grecaux1eqd |
_K1 == _K2 /\ _x1 = _x2 /\ _z1 = _z2 /\ _n1 = _n2 -> grecaux1 _K1 _x1 _z1 _n1 = grecaux1 _K2 _x2 _z2 _n2 |
| 11 |
10 |
exp |
_K1 == _K2 /\ _x1 = _x2 /\ _z1 = _z2 -> _n1 = _n2 -> grecaux1 _K1 _x1 _z1 _n1 = grecaux1 _K2 _x2 _z2 _n2 |
| 12 |
11 |
exp |
_K1 == _K2 /\ _x1 = _x2 -> _z1 = _z2 -> _n1 = _n2 -> grecaux1 _K1 _x1 _z1 _n1 = grecaux1 _K2 _x2 _z2 _n2 |
| 13 |
12 |
exp |
_K1 == _K2 -> _x1 = _x2 -> _z1 = _z2 -> _n1 = _n2 -> grecaux1 _K1 _x1 _z1 _n1 = grecaux1 _K2 _x2 _z2 _n2 |
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)