theorem lowereqd (_G: wff) (_A1 _A2: set):
$ _G -> _A1 == _A2 $ >
$ _G -> lower _A1 = lower _A2 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqsidd |
_G -> n == n |
| 2 |
|
hyp _Ah |
_G -> _A1 == _A2 |
| 3 |
1, 2 |
eqseqd |
_G -> (n == _A1 <-> n == _A2) |
| 4 |
3 |
abeqd |
_G -> {n | n == _A1} == {n | n == _A2} |
| 5 |
4 |
theeqd |
_G -> the {n | n == _A1} = the {n | n == _A2} |
| 6 |
5 |
conv lower |
_G -> lower _A1 = lower _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)