Theorem recnauxeqd ≪ | index | src | ≫

theorem recnauxeqd (_G: wff) (_z1 _z2: nat) (_S1 _S2: set) (_n1 _n2: nat):
  $ _G -> _z1 = _z2 $ >
  $ _G -> _S1 == _S2 $ >
  $ _G -> _n1 = _n2 $ >
  $ _G -> recnaux _z1 _S1 _n1 = recnaux _z2 _S2 _n2 $;


_G -> 0 = 0
_G -> _z1 = _z2
_G -> 0, _z1 = 0, _z2
_G -> suc (fst p) = suc (fst p)
_G -> _S1 == _S2
_G -> p = p
_G -> _S1 @ p = _S2 @ p
_G -> suc (fst p), _S1 @ p = suc (fst p), _S2 @ p
_G -> \ p, suc (fst p), _S1 @ p == \ p, suc (fst p), _S2 @ p
_G -> _n1 = _n2
_G -> rec (0, _z1) (\ p, suc (fst p), _S1 @ p) _n1 = rec (0, _z2) (\ p, suc (fst p), _S2 @ p) _n2
_G -> recnaux _z1 _S1 _n1 = recnaux _z2 _S2 _n2

Step	Hyp	Ref	Expression
1		eqidd	_G -> 0 = 0
2		hyp _zh	_G -> _z1 = _z2
3	1, 2	preqd	_G -> 0, _z1 = 0, _z2
4		eqidd	_G -> suc (fst p) = suc (fst p)
5		hyp _Sh	_G -> _S1 == _S2
6		eqidd	_G -> p = p
7	5, 6	appeqd	_G -> _S1 @ p = _S2 @ p
8	4, 7	preqd	_G -> suc (fst p), _S1 @ p = suc (fst p), _S2 @ p
9	8	lameqd	_G -> \ p, suc (fst p), _S1 @ p == \ p, suc (fst p), _S2 @ p
10		hyp _nh	_G -> _n1 = _n2
11	3, 9, 10	receqd	_G -> rec (0, _z1) (\ p, suc (fst p), _S1 @ p) _n1 = rec (0, _z2) (\ p, suc (fst p), _S2 @ p) _n2
12	11	conv recnaux	_G -> recnaux _z1 _S1 _n1 = recnaux _z2 _S2 _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)