theorem leeqd (_G: wff) (_a1 _a2 _b1 _b2: nat):
$ _G -> _a1 = _a2 $ >
$ _G -> _b1 = _b2 $ >
$ _G -> (_a1 <= _b1 <-> _a2 <= _b2) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
hyp _ah |
_G -> _a1 = _a2 |
| 2 |
|
eqidd |
_G -> x = x |
| 3 |
1, 2 |
addeqd |
_G -> _a1 + x = _a2 + x |
| 4 |
|
hyp _bh |
_G -> _b1 = _b2 |
| 5 |
3, 4 |
eqeqd |
_G -> (_a1 + x = _b1 <-> _a2 + x = _b2) |
| 6 |
5 |
exeqd |
_G -> (E. x _a1 + x = _b1 <-> E. x _a2 + x = _b2) |
| 7 |
6 |
conv le |
_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),
axs_peano
(addeq)