Theorem zdvdeq ≪ | index | src | ≫

theorem zdvdeq (_m1 _m2 _n1 _n2: nat):
  $ _m1 = _m2 -> _n1 = _n2 -> (_m1 |Z _n1 <-> _m2 |Z _n2) $;

Step	Hyp	Ref	Expression
1		anl	_m1 = _m2 /\ _n1 = _n2 -> _m1 = _m2
2		anr	_m1 = _m2 /\ _n1 = _n2 -> _n1 = _n2
3	1, 2	zdvdeqd	_m1 = _m2 /\ _n1 = _n2 -> (_m1 \|Z _n1 <-> _m2 \|Z _n2)
4	3	exp	_m1 = _m2 -> _n1 = _n2 -> (_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)