theorem lfnauxeqd (_G: wff) (_F1 _F2: set) (_k1 _k2 _n1 _n2: nat):
$ _G -> _F1 == _F2 $ >
$ _G -> _k1 = _k2 $ >
$ _G -> _n1 = _n2 $ >
$ _G -> lfnaux _F1 _k1 _n1 = lfnaux _F2 _k2 _n2 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqidd |
_G -> 0 = 0 |
| 2 |
|
eqsidd |
_G -> (\\ _1, \ x, suc x) == (\\ _1, \ x, suc x) |
| 3 |
|
hyp _Fh |
_G -> _F1 == _F2 |
| 4 |
|
eqidd |
_G -> i = i |
| 5 |
3, 4 |
appeqd |
_G -> _F1 @ i = _F2 @ i |
| 6 |
|
eqidd |
_G -> ih = ih |
| 7 |
5, 6 |
conseqd |
_G -> _F1 @ i : ih = _F2 @ i : ih |
| 8 |
7 |
lameqd |
_G -> \ ih, _F1 @ i : ih == \ ih, _F2 @ i : ih |
| 9 |
8 |
slameqd |
_G -> (\\ i, \ ih, _F1 @ i : ih) == (\\ i, \ ih, _F2 @ i : ih) |
| 10 |
9 |
slameqd |
_G -> (\\ _2, \\ i, \ ih, _F1 @ i : ih) == (\\ _2, \\ i, \ ih, _F2 @ i : ih) |
| 11 |
|
hyp _nh |
_G -> _n1 = _n2 |
| 12 |
|
hyp _kh |
_G -> _k1 = _k2 |
| 13 |
1, 2, 10, 11, 12 |
greceqd |
_G -> grec 0 (\\ _1, \ x, suc x) (\\ _2, \\ i, \ ih, _F1 @ i : ih) _n1 _k1 = grec 0 (\\ _1, \ x, suc x) (\\ _2, \\ i, \ ih, _F2 @ i : ih) _n2 _k2 |
| 14 |
13 |
conv lfnaux |
_G -> lfnaux _F1 _k1 _n1 = lfnaux _F2 _k2 _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)