theorem Ifeqd (_G _p1 _p2: wff) (_A1 _A2 _B1 _B2: set):
$ _G -> (_p1 <-> _p2) $ >
$ _G -> _A1 == _A2 $ >
$ _G -> _B1 == _B2 $ >
$ _G -> If _p1 _A1 _B1 == If _p2 _A2 _B2 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
hyp _ph |
_G -> (_p1 <-> _p2) |
| 2 |
|
eqidd |
_G -> n = n |
| 3 |
|
hyp _Ah |
_G -> _A1 == _A2 |
| 4 |
2, 3 |
eleqd |
_G -> (n e. _A1 <-> n e. _A2) |
| 5 |
|
hyp _Bh |
_G -> _B1 == _B2 |
| 6 |
2, 5 |
eleqd |
_G -> (n e. _B1 <-> n e. _B2) |
| 7 |
1, 4, 6 |
ifpeqd |
_G -> (ifp _p1 (n e. _A1) (n e. _B1) <-> ifp _p2 (n e. _A2) (n e. _B2)) |
| 8 |
7 |
abeqd |
_G -> {n | ifp _p1 (n e. _A1) (n e. _B1)} == {n | ifp _p2 (n e. _A2) (n e. _B2)} |
| 9 |
8 |
conv If |
_G -> If _p1 _A1 _B1 == If _p2 _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)