theorem appeqd (_G: wff) (_F1 _F2: set) (_x1 _x2: nat):
$ _G -> _F1 == _F2 $ >
$ _G -> _x1 = _x2 $ >
$ _G -> _F1 @ _x1 = _F2 @ _x2 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
hyp _xh |
_G -> _x1 = _x2 |
| 2 |
|
eqidd |
_G -> y = y |
| 3 |
1, 2 |
preqd |
_G -> _x1, y = _x2, y |
| 4 |
|
hyp _Fh |
_G -> _F1 == _F2 |
| 5 |
3, 4 |
eleqd |
_G -> (_x1, y e. _F1 <-> _x2, y e. _F2) |
| 6 |
5 |
abeqd |
_G -> {y | _x1, y e. _F1} == {y | _x2, y e. _F2} |
| 7 |
6 |
theeqd |
_G -> the {y | _x1, y e. _F1} = the {y | _x2, y e. _F2} |
| 8 |
7 |
conv app |
_G -> _F1 @ _x1 = _F2 @ _x2 |
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)