theorem writeeqd (_G: wff) (_F1 _F2: set) (_a1 _a2 _b1 _b2: nat):
$ _G -> _F1 == _F2 $ >
$ _G -> _a1 = _a2 $ >
$ _G -> _b1 = _b2 $ >
$ _G -> write _F1 _a1 _b1 == write _F2 _a2 _b2 $;
Step | Hyp | Ref | Expression |
1 |
|
eqidd |
_G -> x = x |
2 |
|
hyp _ah |
_G -> _a1 = _a2 |
3 |
1, 2 |
eqeqd |
_G -> (x = _a1 <-> x = _a2) |
4 |
|
eqidd |
_G -> y = y |
5 |
|
hyp _bh |
_G -> _b1 = _b2 |
6 |
4, 5 |
eqeqd |
_G -> (y = _b1 <-> y = _b2) |
7 |
|
eqidd |
_G -> x, y = x, y |
8 |
|
hyp _Fh |
_G -> _F1 == _F2 |
9 |
7, 8 |
eleqd |
_G -> (x, y e. _F1 <-> x, y e. _F2) |
10 |
3, 6, 9 |
ifpeqd |
_G -> (ifp (x = _a1) (y = _b1) (x, y e. _F1) <-> ifp (x = _a2) (y = _b2) (x, y e. _F2)) |
11 |
10 |
abeqd |
_G -> {y | ifp (x = _a1) (y = _b1) (x, y e. _F1)} == {y | ifp (x = _a2) (y = _b2) (x, y e. _F2)} |
12 |
11 |
sabeqd |
_G -> S\ x, {y | ifp (x = _a1) (y = _b1) (x, y e. _F1)} == S\ x, {y | ifp (x = _a2) (y = _b2) (x, y e. _F2)} |
13 |
12 |
conv write |
_G -> write _F1 _a1 _b1 == write _F2 _a2 _b2 |
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)