Theorem elListS ≪ | index | src | ≫ 

theorem elListS (A: set) (a b: nat):
  $ a : b e. List A <-> a e. A /\ b e. List A $;
StepHypRefExpression
1 bitr 
(a : b e. List A <-> all A (a : b)) -> (all A (a : b) <-> a e. A /\ b e. List A) -> (a : b e. List A <-> a e. A /\ b e. List A)
2 elList 
a : b e. List A <-> all A (a : b)
3 1, 2 ax_mp 
(all A (a : b) <-> a e. A /\ b e. List A) -> (a : b e. List A <-> a e. A /\ b e. List A)
4 bitr4 
(all A (a : b) <-> a e. A /\ all A b) -> (a e. A /\ b e. List A <-> a e. A /\ all A b) -> (all A (a : b) <-> a e. A /\ b e. List A)
5 allS 
all A (a : b) <-> a e. A /\ all A b
6 4, 5 ax_mp 
(a e. A /\ b e. List A <-> a e. A /\ all A b) -> (all A (a : b) <-> a e. A /\ b e. List A)
7 elList 
b e. List A <-> all A b
8 7 aneq2i 
a e. A /\ b e. List A <-> a e. A /\ all A b
9 6, 8 ax_mp 
all A (a : b) <-> a e. A /\ b e. List A
10 3, 9 ax_mp 
a : b e. List A <-> a e. 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)