theorem srecauxeqd (_G: wff) (_S1 _S2: set) (_n1 _n2: nat):
$ _G -> _S1 == _S2 $ >
$ _G -> _n1 = _n2 $ >
$ _G -> srecaux _S1 _n1 = srecaux _S2 _n2 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqidd |
_G -> 0 = 0 |
| 2 |
|
hyp _Sh |
_G -> _S1 == _S2 |
| 3 |
2 |
appeq1d |
_G -> _S1 @ b = _S2 @ b |
| 4 |
3 |
writeeq3d |
_G -> write b a (_S1 @ b) == write b a (_S2 @ b) |
| 5 |
4 |
lowereqd |
_G -> lower (write b a (_S1 @ b)) = lower (write b a (_S2 @ b)) |
| 6 |
5 |
lameqd |
_G -> \ b, lower (write b a (_S1 @ b)) == \ b, lower (write b a (_S2 @ b)) |
| 7 |
6 |
slameqd |
_G -> (\\ a, \ b, lower (write b a (_S1 @ b))) == (\\ a, \ b, lower (write b a (_S2 @ b))) |
| 8 |
|
hyp _nh |
_G -> _n1 = _n2 |
| 9 |
1, 7, 8 |
recneqd |
_G -> recn 0 (\\ a, \ b, lower (write b a (_S1 @ b))) _n1 = recn 0 (\\ a, \ b, lower (write b a (_S2 @ b))) _n2 |
| 10 |
9 |
conv srecaux |
_G -> srecaux _S1 _n1 = srecaux _S2 _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)