Theorem sublistAteqd ≪ | index | src | ≫ 

theorem sublistAteqd (_G: wff) (_n1 _n2 _L11 _L12 _L21 _L22: nat):
  $ _G -> _n1 = _n2 $ >
  $ _G -> _L11 = _L12 $ >
  $ _G -> _L21 = _L22 $ >
  $ _G -> (sublistAt _n1 _L11 _L21 <-> sublistAt _n2 _L12 _L22) $;
StepHypRefExpression
1 hyp _L1h 
_G -> _L11 = _L12
2 eqidd 
_G -> l = l
3 hyp _L2h 
_G -> _L21 = _L22
4 eqidd 
_G -> r = r
5 3, 4 appendeqd 
_G -> _L21 ++ r = _L22 ++ r
6 2, 5 appendeqd 
_G -> l ++ _L21 ++ r = l ++ _L22 ++ r
7 1, 6 eqeqd 
_G -> (_L11 = l ++ _L21 ++ r <-> _L12 = l ++ _L22 ++ r)
8 eqidd 
_G -> len l = len l
9 hyp _nh 
_G -> _n1 = _n2
10 8, 9 eqeqd 
_G -> (len l = _n1 <-> len l = _n2)
11 7, 10 aneqd 
_G -> (_L11 = l ++ _L21 ++ r /\ len l = _n1 <-> _L12 = l ++ _L22 ++ r /\ len l = _n2)
12 11 exeqd 
_G -> (E. r (_L11 = l ++ _L21 ++ r /\ len l = _n1) <-> E. r (_L12 = l ++ _L22 ++ r /\ len l = _n2))
13 12 exeqd 
_G -> (E. l E. r (_L11 = l ++ _L21 ++ r /\ len l = _n1) <-> E. l E. r (_L12 = l ++ _L22 ++ r /\ len l = _n2))
14 13 conv sublistAt 
_G -> (sublistAt _n1 _L11 _L21 <-> sublistAt _n2 _L12 _L22)

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)