theorem coprimeeq (_a1 _a2 _b1 _b2: nat):
$ _a1 = _a2 -> _b1 = _b2 -> (coprime _a1 _b1 <-> coprime _a2 _b2) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
anl |
_a1 = _a2 /\ _b1 = _b2 -> _a1 = _a2 |
| 2 |
|
anr |
_a1 = _a2 /\ _b1 = _b2 -> _b1 = _b2 |
| 3 |
1, 2 |
coprimeeqd |
_a1 = _a2 /\ _b1 = _b2 -> (coprime _a1 _b1 <-> coprime _a2 _b2) |
| 4 |
3 |
exp |
_a1 = _a2 -> _b1 = _b2 -> (coprime _a1 _b1 <-> coprime _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
(muleq)