theorem subsneqd (_G: wff) (_A1 _A2: set):
$ _G -> _A1 == _A2 $ >
$ _G -> (subsn _A1 <-> subsn _A2) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
hyp _Ah |
_G -> _A1 == _A2 |
| 2 |
1 |
eleq2d |
_G -> (x e. _A1 <-> x e. _A2) |
| 3 |
1 |
eleq2d |
_G -> (y e. _A1 <-> y e. _A2) |
| 4 |
3 |
imeq1d |
_G -> (y e. _A1 -> x = y <-> y e. _A2 -> x = y) |
| 5 |
2, 4 |
imeqd |
_G -> (x e. _A1 -> y e. _A1 -> x = y <-> x e. _A2 -> y e. _A2 -> x = y) |
| 6 |
5 |
aleqd |
_G -> (A. y (x e. _A1 -> y e. _A1 -> x = y) <-> A. y (x e. _A2 -> y e. _A2 -> x = y)) |
| 7 |
6 |
aleqd |
_G -> (A. x A. y (x e. _A1 -> y e. _A1 -> x = y) <-> A. x A. y (x e. _A2 -> y e. _A2 -> x = y)) |
| 8 |
7 |
conv subsn |
_G -> (subsn _A1 <-> subsn _A2) |
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)