theorem uneqd (_G: wff) (_A1 _A2 _B1 _B2: set):
$ _G -> _A1 == _A2 $ >
$ _G -> _B1 == _B2 $ >
$ _G -> _A1 u. _B1 == _A2 u. _B2 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqidd |
_G -> x = x |
| 2 |
|
hyp _Ah |
_G -> _A1 == _A2 |
| 3 |
1, 2 |
eleqd |
_G -> (x e. _A1 <-> x e. _A2) |
| 4 |
|
hyp _Bh |
_G -> _B1 == _B2 |
| 5 |
1, 4 |
eleqd |
_G -> (x e. _B1 <-> x e. _B2) |
| 6 |
3, 5 |
oreqd |
_G -> (x e. _A1 \/ x e. _B1 <-> x e. _A2 \/ x e. _B2) |
| 7 |
6 |
abeqd |
_G -> {x | x e. _A1 \/ x e. _B1} == {x | x e. _A2 \/ x e. _B2} |
| 8 |
7 |
conv Union |
_G -> _A1 u. _B1 == _A2 u. _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)