Theorem appendT ≪ | index | src | ≫ 

theorem appendT (A: set) (a b: nat):
  $ a ++ b e. List A <-> a e. List A /\ b e. List A $;
StepHypRefExpression
1 bitr4 
(a ++ b e. List A <-> all A (a ++ b)) -> (a e. List A /\ b e. List A <-> all A (a ++ b)) -> (a ++ b e. List A <-> a e. List A /\ b e. List A)
2 elList 
a ++ b e. List A <-> all A (a ++ b)
3 1, 2 ax_mp 
(a e. List A /\ b e. List A <-> all A (a ++ b)) -> (a ++ b e. List A <-> a e. List A /\ b e. List A)
4 bitr4 
(a e. List A /\ b e. List A <-> all A a /\ all A b) -> (all A (a ++ b) <-> all A a /\ all A b) -> (a e. List A /\ b e. List A <-> all A (a ++ b))
5 aneq 
(a e. List A <-> all A a) -> (b e. List A <-> all A b) -> (a e. List A /\ b e. List A <-> all A a /\ all A b)
6 elList 
a e. List A <-> all A a
7 5, 6 ax_mp 
(b e. List A <-> all A b) -> (a e. List A /\ b e. List A <-> all A a /\ all A b)
8 elList 
b e. List A <-> all A b
9 7, 8 ax_mp 
a e. List A /\ b e. List A <-> all A a /\ all A b
10 4, 9 ax_mp 
(all A (a ++ b) <-> all A a /\ all A b) -> (a e. List A /\ b e. List A <-> all A (a ++ b))
11 allappend 
all A (a ++ b) <-> all A a /\ all A b
12 10, 11 ax_mp 
a e. List A /\ b e. List A <-> all A (a ++ b)
13 3, 12 ax_mp 
a ++ b e. List A <-> a e. List A /\ b e. List A

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)