Theorem Arrowsn2 ≪ | index | src | ≫ 

theorem Arrowsn2 (a b: nat) {x: nat}: $ Arrow a (sn b) == sn (\. x e. a, b) $;
StepHypRefExpression
1 bitr4 
(a1 e. Arrow a (sn b) <-> func a1 a (sn b)) -> (a1 e. sn (\. x e. a, b) <-> func a1 a (sn b)) -> (a1 e. Arrow a (sn b) <-> a1 e. sn (\. x e. a, b))
2 elArrow 
a1 e. Arrow a (sn b) <-> func a1 a (sn b)
3 1, 2 ax_mp 
(a1 e. sn (\. x e. a, b) <-> func a1 a (sn b)) -> (a1 e. Arrow a (sn b) <-> a1 e. sn (\. x e. a, b))
4 bitr4 
(a1 e. sn (\. x e. a, b) <-> a1 = \. x e. a, b) -> (func a1 a (sn b) <-> a1 = \. x e. a, b) -> (a1 e. sn (\. x e. a, b) <-> func a1 a (sn b))
5 elsn 
a1 e. sn (\. x e. a, b) <-> a1 = \. x e. a, b
6 4, 5 ax_mp 
(func a1 a (sn b) <-> a1 = \. x e. a, b) -> (a1 e. sn (\. x e. a, b) <-> func a1 a (sn b))
7 bitr 
(func a1 a (sn b) <-> a1 == \. x e. a, b) -> (a1 == \. x e. a, b <-> a1 = \. x e. a, b) -> (func a1 a (sn b) <-> a1 = \. x e. a, b)
8 funcsn2 
func a1 a (sn b) <-> a1 == \. x e. a, b
9 7, 8 ax_mp 
(a1 == \. x e. a, b <-> a1 = \. x e. a, b) -> (func a1 a (sn b) <-> a1 = \. x e. a, b)
10 nsinj 
a1 == \. x e. a, b <-> a1 = \. x e. a, b
11 9, 10 ax_mp 
func a1 a (sn b) <-> a1 = \. x e. a, b
12 6, 11 ax_mp 
a1 e. sn (\. x e. a, b) <-> func a1 a (sn b)
13 3, 12 ax_mp 
a1 e. Arrow a (sn b) <-> a1 e. sn (\. x e. a, b)
14 13 ax_gen 
A. a1 (a1 e. Arrow a (sn b) <-> a1 e. sn (\. x e. a, b))
15 14 conv eqs 
Arrow a (sn b) == sn (\. x e. a, b)

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 (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)