theorem alleqd (_G: wff) (_A1 _A2: set) (_l1 _l2: nat):
$ _G -> _A1 == _A2 $ >
$ _G -> _l1 = _l2 $ >
$ _G -> (all _A1 _l1 <-> all _A2 _l2) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
hyp _lh |
_G -> _l1 = _l2 |
| 2 |
1 |
lmemseqd |
_G -> lmems _l1 = lmems _l2 |
| 3 |
2 |
nseqd |
_G -> lmems _l1 == lmems _l2 |
| 4 |
|
hyp _Ah |
_G -> _A1 == _A2 |
| 5 |
3, 4 |
sseqd |
_G -> (lmems _l1 C_ _A1 <-> lmems _l2 C_ _A2) |
| 6 |
5 |
conv all |
_G -> (all _A1 _l1 <-> all _A2 _l2) |
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)