theorem eqseqd (_G: wff) (_A1 _A2 _B1 _B2: set):
$ _G -> _A1 == _A2 $ >
$ _G -> _B1 == _B2 $ >
$ _G -> (_A1 == _B1 <-> _A2 == _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 |
bieqd |
_G -> (x e. _A1 <-> x e. _B1 <-> (x e. _A2 <-> x e. _B2)) |
7 |
6 |
aleqd |
_G -> (A. x (x e. _A1 <-> x e. _B1) <-> A. x (x e. _A2 <-> x e. _B2)) |
8 |
7 |
conv eqs |
_G -> (_A1 == _B1 <-> _A2 == _B2) |
Axiom use
axs_prop_calc
(ax_1,
ax_2,
ax_3,
ax_mp),
axs_pred_calc
(ax_gen,
ax_4,
ax_5,
ax_6,
ax_7,
ax_10,
ax_12),
axs_set
(ax_8)