Theorem elall22 ≪ | index | src | ≫ 

theorem elall22 (R: set) (l1 l2: nat) {n x y: nat}:
  $ l1, l2 e. all2 R <->
    len l1 = len l2 /\
      A. n A. x (nth n l1 = suc x -> A. y (nth n l2 = suc y -> x, y e. R)) $;
StepHypRefExpression
1 bitr 
(l1, l2 e. all2 R <-> len l1 = len l2 /\ A. n A. x A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. R)) ->
  (len l1 = len l2 /\ A. n A. x A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. R) <->
    len l1 = len l2 /\ A. n A. x (nth n l1 = suc x -> A. y (nth n l2 = suc y -> x, y e. R))) ->
  (l1, l2 e. all2 R <-> len l1 = len l2 /\ A. n A. x (nth n l1 = suc x -> A. y (nth n l2 = suc y -> x, y e. R)))
2 elall2 
l1, l2 e. all2 R <-> len l1 = len l2 /\ A. n A. x A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. R)
3 1, 2 ax_mp 
(len l1 = len l2 /\ A. n A. x A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. R) <->
    len l1 = len l2 /\ A. n A. x (nth n l1 = suc x -> A. y (nth n l2 = suc y -> x, y e. R))) ->
  (l1, l2 e. all2 R <-> len l1 = len l2 /\ A. n A. x (nth n l1 = suc x -> A. y (nth n l2 = suc y -> x, y e. R)))
4 alim1 
A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. R) <-> nth n l1 = suc x -> A. y (nth n l2 = suc y -> x, y e. R)
5 4 aleqi 
A. x A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. R) <-> A. x (nth n l1 = suc x -> A. y (nth n l2 = suc y -> x, y e. R))
6 5 aleqi 
A. n A. x A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. R) <-> A. n A. x (nth n l1 = suc x -> A. y (nth n l2 = suc y -> x, y e. R))
7 6 aneq2i 
len l1 = len l2 /\ A. n A. x A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. R) <->
  len l1 = len l2 /\ A. n A. x (nth n l1 = suc x -> A. y (nth n l2 = suc y -> x, y e. R))
8 3, 7 ax_mp 
l1, l2 e. all2 R <-> len l1 = len l2 /\ A. n A. x (nth n l1 = suc x -> A. y (nth n l2 = suc y -> x, y e. R))

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)