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) $;

Step	Hyp	Ref	Expression
1		hyp _L1h	_G -> _L11 = _L12
2		hyp _L2h	_G -> _L21 = _L22
3	2	appendeq1d	_G -> _L21 ++ r = _L22 ++ r
4	3	appendeq2d	_G -> l ++ _L21 ++ r = l ++ _L22 ++ r
5	1, 4	eqeqd	_G -> (_L11 = l ++ _L21 ++ r <-> _L12 = l ++ _L22 ++ r)
6		hyp _nh	_G -> _n1 = _n2
7	6	eqeq2d	_G -> (len l = _n1 <-> len l = _n2)
8	5, 7	aneqd	_G -> (_L11 = l ++ _L21 ++ r /\ len l = _n1 <-> _L12 = l ++ _L22 ++ r /\ len l = _n2)
9	8	exeqd	_G -> (E. r (_L11 = l ++ _L21 ++ r /\ len l = _n1) <-> E. r (_L12 = l ++ _L22 ++ r /\ len l = _n2))
10	9	exeqd	_G -> (E. l E. r (_L11 = l ++ _L21 ++ r /\ len l = _n1) <-> E. l E. r (_L12 = l ++ _L22 ++ r /\ len l = _n2))
11	10	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)