Theorem eqrappb ≪ | index | src | ≫ 

theorem eqrappb (A B: set) {x: nat}: $ A == B <-> A. x A @' x == B @' x $;
StepHypRefExpression
1 bitr4 
(A == B <-> A C_ B /\ B C_ A) -> (A. x A @' x == B @' x <-> A C_ B /\ B C_ A) -> (A == B <-> A. x A @' x == B @' x)
2 ssasymb 
A == B <-> A C_ B /\ B C_ A
3 1, 2 ax_mp 
(A. x A @' x == B @' x <-> A C_ B /\ B C_ A) -> (A == B <-> A. x A @' x == B @' x)
4 bitr 
(A. x A @' x == B @' x <-> A. x (A @' x C_ B @' x /\ B @' x C_ A @' x)) ->
  (A. x (A @' x C_ B @' x /\ B @' x C_ A @' x) <-> A C_ B /\ B C_ A) ->
  (A. x A @' x == B @' x <-> A C_ B /\ B C_ A)
5 ssasymb 
A @' x == B @' x <-> A @' x C_ B @' x /\ B @' x C_ A @' x
6 5 aleqi 
A. x A @' x == B @' x <-> A. x (A @' x C_ B @' x /\ B @' x C_ A @' x)
7 4, 6 ax_mp 
(A. x (A @' x C_ B @' x /\ B @' x C_ A @' x) <-> A C_ B /\ B C_ A) -> (A. x A @' x == B @' x <-> A C_ B /\ B C_ A)
8 bitr4 
(A. x (A @' x C_ B @' x /\ B @' x C_ A @' x) <-> A. x A @' x C_ B @' x /\ A. x B @' x C_ A @' x) ->
  (A C_ B /\ B C_ A <-> A. x A @' x C_ B @' x /\ A. x B @' x C_ A @' x) ->
  (A. x (A @' x C_ B @' x /\ B @' x C_ A @' x) <-> A C_ B /\ B C_ A)
9 alan 
A. x (A @' x C_ B @' x /\ B @' x C_ A @' x) <-> A. x A @' x C_ B @' x /\ A. x B @' x C_ A @' x
10 8, 9 ax_mp 
(A C_ B /\ B C_ A <-> A. x A @' x C_ B @' x /\ A. x B @' x C_ A @' x) -> (A. x (A @' x C_ B @' x /\ B @' x C_ A @' x) <-> A C_ B /\ B C_ A)
11 aneq 
(A C_ B <-> A. x A @' x C_ B @' x) -> (B C_ A <-> A. x B @' x C_ A @' x) -> (A C_ B /\ B C_ A <-> A. x A @' x C_ B @' x /\ A. x B @' x C_ A @' x)
12 rappssb 
A C_ B <-> A. x A @' x C_ B @' x
13 11, 12 ax_mp 
(B C_ A <-> A. x B @' x C_ A @' x) -> (A C_ B /\ B C_ A <-> A. x A @' x C_ B @' x /\ A. x B @' x C_ A @' x)
14 rappssb 
B C_ A <-> A. x B @' x C_ A @' x
15 13, 14 ax_mp 
A C_ B /\ B C_ A <-> A. x A @' x C_ B @' x /\ A. x B @' x C_ A @' x
16 10, 15 ax_mp 
A. x (A @' x C_ B @' x /\ B @' x C_ A @' x) <-> A C_ B /\ B C_ A
17 7, 16 ax_mp 
A. x A @' x == B @' x <-> A C_ B /\ B C_ A
18 3, 17 ax_mp 
A == B <-> A. x A @' x == B @' x

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)