Theorem ex2eqd ≪ | index | src | ≫ 

theorem ex2eqd (_G: wff) (_R1 _R2: set):
  $ _G -> _R1 == _R2 $ >
  $ _G -> ex2 _R1 == ex2 _R2 $;
StepHypRefExpression
1 biidd 
_G -> (len l1 = len l2 <-> len l1 = len l2)
2 biidd 
_G -> (nth n l1 = suc x /\ nth n l2 = suc y <-> nth n l1 = suc x /\ nth n l2 = suc y)
3 eqidd 
_G -> x, y = x, y
4 hyp _Rh 
_G -> _R1 == _R2
5 3, 4 eleqd 
_G -> (x, y e. _R1 <-> x, y e. _R2)
6 2, 5 aneqd 
_G -> (nth n l1 = suc x /\ nth n l2 = suc y /\ x, y e. _R1 <-> nth n l1 = suc x /\ nth n l2 = suc y /\ x, y e. _R2)
7 6 exeqd 
_G -> (E. y (nth n l1 = suc x /\ nth n l2 = suc y /\ x, y e. _R1) <-> E. y (nth n l1 = suc x /\ nth n l2 = suc y /\ x, y e. _R2))
8 7 exeqd 
_G -> (E. x E. y (nth n l1 = suc x /\ nth n l2 = suc y /\ x, y e. _R1) <-> E. x E. y (nth n l1 = suc x /\ nth n l2 = suc y /\ x, y e. _R2))
9 8 exeqd 
_G -> (E. n E. x E. y (nth n l1 = suc x /\ nth n l2 = suc y /\ x, y e. _R1) <-> E. n E. x E. y (nth n l1 = suc x /\ nth n l2 = suc y /\ x, y e. _R2))
10 1, 9 aneqd 
_G ->
  (len l1 = len l2 /\ E. n E. x E. y (nth n l1 = suc x /\ nth n l2 = suc y /\ x, y e. _R1) <->
    len l1 = len l2 /\ E. n E. x E. y (nth n l1 = suc x /\ nth n l2 = suc y /\ x, y e. _R2))
11 10 abeqd 
_G ->
  {l2 | len l1 = len l2 /\ E. n E. x E. y (nth n l1 = suc x /\ nth n l2 = suc y /\ x, y e. _R1)} ==
    {l2 | len l1 = len l2 /\ E. n E. x E. y (nth n l1 = suc x /\ nth n l2 = suc y /\ x, y e. _R2)}
12 11 sabeqd 
_G ->
  S\ l1, {l2 | len l1 = len l2 /\ E. n E. x E. y (nth n l1 = suc x /\ nth n l2 = suc y /\ x, y e. _R1)} ==
    S\ l1, {l2 | len l1 = len l2 /\ E. n E. x E. y (nth n l1 = suc x /\ nth n l2 = suc y /\ x, y e. _R2)}
13 12 conv ex2 
_G -> ex2 _R1 == ex2 _R2

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)