Theorem zeqmeq | index | src |

theorem zeqmeq (_n1 _n2 _a1 _a2 _b1 _b2: nat):
  $ _n1 = _n2 ->
    _a1 = _a2 ->
    _b1 = _b2 ->
    (modZ(_n1): _a1 = _b1 <-> modZ(_n2): _a2 = _b2) $;
StepHypRefExpression
1 anl
_n1 = _n2 /\ _a1 = _a2 -> _n1 = _n2
2 1 anwl
_n1 = _n2 /\ _a1 = _a2 /\ _b1 = _b2 -> _n1 = _n2
3 anr
_n1 = _n2 /\ _a1 = _a2 -> _a1 = _a2
4 3 anwl
_n1 = _n2 /\ _a1 = _a2 /\ _b1 = _b2 -> _a1 = _a2
5 anr
_n1 = _n2 /\ _a1 = _a2 /\ _b1 = _b2 -> _b1 = _b2
6 2, 4, 5 zeqmeqd
_n1 = _n2 /\ _a1 = _a2 /\ _b1 = _b2 -> (modZ(_n1): _a1 = _b1 <-> modZ(_n2): _a2 = _b2)
7 6 exp
_n1 = _n2 /\ _a1 = _a2 -> _b1 = _b2 -> (modZ(_n1): _a1 = _b1 <-> modZ(_n2): _a2 = _b2)
8 7 exp
_n1 = _n2 -> _a1 = _a2 -> _b1 = _b2 -> (modZ(_n1): _a1 = _b1 <-> modZ(_n2): _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 (peano2, addeq, muleq)