theorem sabeqd (_G: wff) {x: nat} (_A1 _A2: set x):
$ _G -> _A1 == _A2 $ >
$ _G -> S\ x, _A1 == S\ x, _A2 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqidd |
_G -> snd z = snd z |
| 2 |
|
eqidd |
_G -> fst z = fst z |
| 3 |
|
hyp _Ah |
_G -> _A1 == _A2 |
| 4 |
2, 3 |
sbseqd |
_G -> S[fst z / x] _A1 == S[fst z / x] _A2 |
| 5 |
1, 4 |
eleqd |
_G -> (snd z e. S[fst z / x] _A1 <-> snd z e. S[fst z / x] _A2) |
| 6 |
5 |
abeqd |
_G -> {z | snd z e. S[fst z / x] _A1} == {z | snd z e. S[fst z / x] _A2} |
| 7 |
6 |
conv sab |
_G -> S\ x, _A1 == S\ x, _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)