Theorem receqd ≪ | index | src | ≫

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

Step	Hyp	Ref	Expression
1		hyp _zh	_G -> _z1 = _z2
2	1	eqeq2d	_G -> (pset a @ 0 = _z1 <-> pset a @ 0 = _z2)
3		hyp _nh	_G -> _n1 = _n2
4	3	appeq2d	_G -> pset a @ _n1 = pset a @ _n2
5	4	eqeq1d	_G -> (pset a @ _n1 = v <-> pset a @ _n2 = v)
6	2, 5	aneqd	_G -> (pset a @ 0 = _z1 /\ pset a @ _n1 = v <-> pset a @ 0 = _z2 /\ pset a @ _n2 = v)
7	3	lteq2d	_G -> (i < _n1 <-> i < _n2)
8		hyp _Sh	_G -> _S1 == _S2
9	8	appeq1d	_G -> _S1 @ (pset a @ i) = _S2 @ (pset a @ i)
10	9	eqeq2d	_G -> (pset a @ suc i = _S1 @ (pset a @ i) <-> pset a @ suc i = _S2 @ (pset a @ i))
11	7, 10	imeqd	_G -> (i < _n1 -> pset a @ suc i = _S1 @ (pset a @ i) <-> i < _n2 -> pset a @ suc i = _S2 @ (pset a @ i))
12	11	aleqd	_G -> (A. i (i < _n1 -> pset a @ suc i = _S1 @ (pset a @ i)) <-> A. i (i < _n2 -> pset a @ suc i = _S2 @ (pset a @ i)))
13	6, 12	aneqd	_G -> (pset a @ 0 = _z1 /\ pset a @ _n1 = v /\ A. i (i < _n1 -> pset a @ suc i = _S1 @ (pset a @ i)) <-> pset a @ 0 = _z2 /\ pset a @ _n2 = v /\ A. i (i < _n2 -> pset a @ suc i = _S2 @ (pset a @ i)))
14	13	exeqd	_G -> (E. a (pset a @ 0 = _z1 /\ pset a @ _n1 = v /\ A. i (i < _n1 -> pset a @ suc i = _S1 @ (pset a @ i))) <-> E. a (pset a @ 0 = _z2 /\ pset a @ _n2 = v /\ A. i (i < _n2 -> pset a @ suc i = _S2 @ (pset a @ i))))
15	14	abeqd	_G -> {v \| E. a (pset a @ 0 = _z1 /\ pset a @ _n1 = v /\ A. i (i < _n1 -> pset a @ suc i = _S1 @ (pset a @ i)))} == {v \| E. a (pset a @ 0 = _z2 /\ pset a @ _n2 = v /\ A. i (i < _n2 -> pset a @ suc i = _S2 @ (pset a @ i)))}
16	15	theeqd	_G -> the {v \| E. a (pset a @ 0 = _z1 /\ pset a @ _n1 = v /\ A. i (i < _n1 -> pset a @ suc i = _S1 @ (pset a @ i)))} = the {v \| E. a (pset a @ 0 = _z2 /\ pset a @ _n2 = v /\ A. i (i < _n2 -> pset a @ suc i = _S2 @ (pset a @ i)))}
17	16	conv rec	_G -> rec _z1 _S1 _n1 = rec _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)