theorem sepeqd (_G: wff) (_n1 _n2: nat) (_A1 _A2: set):
$ _G -> _n1 = _n2 $ >
$ _G -> _A1 == _A2 $ >
$ _G -> sep _n1 _A1 = sep _n2 _A2 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
hyp _nh |
_G -> _n1 = _n2 |
| 2 |
1 |
nseqd |
_G -> _n1 == _n2 |
| 3 |
|
hyp _Ah |
_G -> _A1 == _A2 |
| 4 |
2, 3 |
ineqd |
_G -> _n1 i^i _A1 == _n2 i^i _A2 |
| 5 |
4 |
lowereqd |
_G -> lower (_n1 i^i _A1) = lower (_n2 i^i _A2) |
| 6 |
5 |
conv sep |
_G -> sep _n1 _A1 = sep _n2 _A2 |
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)