theorem all2rev (R: set) (l1 l2: nat):
$ rev l1, rev l2 e. all2 R <-> l1, l2 e. all2 R $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eleq1 |
rev (rev l1), rev (rev l2) = l1, l2 -> (rev (rev l1), rev (rev l2) e. all2 R <-> l1, l2 e. all2 R) |
| 2 |
|
preq |
rev (rev l1) = l1 -> rev (rev l2) = l2 -> rev (rev l1), rev (rev l2) = l1, l2 |
| 3 |
|
revrev |
rev (rev l1) = l1 |
| 4 |
2, 3 |
ax_mp |
rev (rev l2) = l2 -> rev (rev l1), rev (rev l2) = l1, l2 |
| 5 |
|
revrev |
rev (rev l2) = l2 |
| 6 |
4, 5 |
ax_mp |
rev (rev l1), rev (rev l2) = l1, l2 |
| 7 |
1, 6 |
ax_mp |
rev (rev l1), rev (rev l2) e. all2 R <-> l1, l2 e. all2 R |
| 8 |
|
all2rev1 |
rev l1, rev l2 e. all2 R -> rev (rev l1), rev (rev l2) e. all2 R |
| 9 |
7, 8 |
sylib |
rev l1, rev l2 e. all2 R -> l1, l2 e. all2 R |
| 10 |
|
all2rev1 |
l1, l2 e. all2 R -> rev l1, rev l2 e. all2 R |
| 11 |
9, 10 |
ibii |
rev l1, rev l2 e. all2 R <-> l1, l2 e. all2 R |
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
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)