theorem zdvdeqd (_G: wff) (_m1 _m2 _n1 _n2: nat):
$ _G -> _m1 = _m2 $ >
$ _G -> _n1 = _n2 $ >
$ _G -> (_m1 |Z _n1 <-> _m2 |Z _n2) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
hyp _mh |
_G -> _m1 = _m2 |
| 2 |
1 |
zabseqd |
_G -> zabs _m1 = zabs _m2 |
| 3 |
|
hyp _nh |
_G -> _n1 = _n2 |
| 4 |
3 |
zabseqd |
_G -> zabs _n1 = zabs _n2 |
| 5 |
2, 4 |
dvdeqd |
_G -> (zabs _m1 || zabs _n1 <-> zabs _m2 || zabs _n2) |
| 6 |
5 |
conv zdvd |
_G -> (_m1 |Z _n1 <-> _m2 |Z _n2) |
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
(peano2,
addeq,
muleq)