theorem sndeqd (_G: wff) (_a1 _a2: nat):
$ _G -> _a1 = _a2 $ >
$ _G -> snd _a1 = snd _a2 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
hyp _ah |
_G -> _a1 = _a2 |
| 2 |
|
eqidd |
_G -> x, y = x, y |
| 3 |
1, 2 |
eqeqd |
_G -> (_a1 = x, y <-> _a2 = x, y) |
| 4 |
3 |
exeqd |
_G -> (E. x _a1 = x, y <-> E. x _a2 = x, y) |
| 5 |
4 |
abeqd |
_G -> {y | E. x _a1 = x, y} == {y | E. x _a2 = x, y} |
| 6 |
5 |
theeqd |
_G -> the {y | E. x _a1 = x, y} = the {y | E. x _a2 = x, y} |
| 7 |
6 |
conv snd |
_G -> snd _a1 = snd _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)