theorem sublistAteq (_n1 _n2 _L11 _L12 _L21 _L22: nat):
$ _n1 = _n2 ->
_L11 = _L12 ->
_L21 = _L22 ->
(sublistAt _n1 _L11 _L21 <-> sublistAt _n2 _L12 _L22) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
anl |
_n1 = _n2 /\ _L11 = _L12 -> _n1 = _n2 |
| 2 |
1 |
anwl |
_n1 = _n2 /\ _L11 = _L12 /\ _L21 = _L22 -> _n1 = _n2 |
| 3 |
|
anr |
_n1 = _n2 /\ _L11 = _L12 -> _L11 = _L12 |
| 4 |
3 |
anwl |
_n1 = _n2 /\ _L11 = _L12 /\ _L21 = _L22 -> _L11 = _L12 |
| 5 |
|
anr |
_n1 = _n2 /\ _L11 = _L12 /\ _L21 = _L22 -> _L21 = _L22 |
| 6 |
2, 4, 5 |
sublistAteqd |
_n1 = _n2 /\ _L11 = _L12 /\ _L21 = _L22 -> (sublistAt _n1 _L11 _L21 <-> sublistAt _n2 _L12 _L22) |
| 7 |
6 |
exp |
_n1 = _n2 /\ _L11 = _L12 -> _L21 = _L22 -> (sublistAt _n1 _L11 _L21 <-> sublistAt _n2 _L12 _L22) |
| 8 |
7 |
exp |
_n1 = _n2 -> _L11 = _L12 -> _L21 = _L22 -> (sublistAt _n1 _L11 _L21 <-> sublistAt _n2 _L12 _L22) |
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)