Theorem srecpeqd ≪ | index | src | ≫

theorem srecpeqd (_G: wff) (_A1 _A2: set) (_n1 _n2: nat):
  $ _G -> _A1 == _A2 $ >
  $ _G -> _n1 = _n2 $ >
  $ _G -> (srecp _A1 _n1 <-> srecp _A2 _n2) $;


_G -> _A1 == _A2
_G -> _n1 = _n2
_G -> srecpaux _A1 _n1 = srecpaux _A2 _n2
_G -> (true (srecpaux _A1 _n1) <-> true (srecpaux _A2 _n2))
_G -> (srecp _A1 _n1 <-> srecp _A2 _n2)

Step	Hyp	Ref	Expression
1		hyp _Ah	_G -> _A1 == _A2
2		hyp _nh	_G -> _n1 = _n2
3	1, 2	srecpauxeqd	_G -> srecpaux _A1 _n1 = srecpaux _A2 _n2
4	3	trueeqd	_G -> (true (srecpaux _A1 _n1) <-> true (srecpaux _A2 _n2))
5	4	conv srecp	_G -> (srecp _A1 _n1 <-> srecp _A2 _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)