Theorem sreceqd ≪ | index | src | ≫

theorem sreceqd (_G: wff) (_S1 _S2: set) (_n1 _n2: nat):
  $ _G -> _S1 == _S2 $ >
  $ _G -> _n1 = _n2 $ >
  $ _G -> srec _S1 _n1 = srec _S2 _n2 $;


_G -> _S1 == _S2
_G -> _n1 = _n2
_G -> suc _n1 = suc _n2
_G -> srecaux _S1 (suc _n1) = srecaux _S2 (suc _n2)
_G -> srecaux _S1 (suc _n1) == srecaux _S2 (suc _n2)
_G -> srecaux _S1 (suc _n1) @ _n1 = srecaux _S2 (suc _n2) @ _n2
_G -> srec _S1 _n1 = srec _S2 _n2

Step	Hyp	Ref	Expression
1		hyp _Sh	_G -> _S1 == _S2
2		hyp _nh	_G -> _n1 = _n2
3	2	suceqd	_G -> suc _n1 = suc _n2
4	1, 3	srecauxeqd	_G -> srecaux _S1 (suc _n1) = srecaux _S2 (suc _n2)
5	4	nseqd	_G -> srecaux _S1 (suc _n1) == srecaux _S2 (suc _n2)
6	5, 2	appeqd	_G -> srecaux _S1 (suc _n1) @ _n1 = srecaux _S2 (suc _n2) @ _n2
7	6	conv srec	_G -> srec _S1 _n1 = srec _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)