theorem subeqd (_G: wff) (_a1 _a2 _b1 _b2: nat):
$ _G -> _a1 = _a2 $ >
$ _G -> _b1 = _b2 $ >
$ _G -> _a1 - _b1 = _a2 - _b2 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
hyp _bh |
_G -> _b1 = _b2 |
| 2 |
|
eqidd |
_G -> x = x |
| 3 |
1, 2 |
addeqd |
_G -> _b1 + x = _b2 + x |
| 4 |
|
hyp _ah |
_G -> _a1 = _a2 |
| 5 |
3, 4 |
eqeqd |
_G -> (_b1 + x = _a1 <-> _b2 + x = _a2) |
| 6 |
5 |
abeqd |
_G -> {x | _b1 + x = _a1} == {x | _b2 + x = _a2} |
| 7 |
6 |
theeqd |
_G -> the {x | _b1 + x = _a1} = the {x | _b2 + x = _a2} |
| 8 |
7 |
conv sub |
_G -> _a1 - _b1 = _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),
axs_the
(theid,
the0),
axs_peano
(addeq)