Theorem all2eqd ≪ | index | src | ≫ 

theorem all2eqd (_G: wff) (_R1 _R2: set):
  $ _G -> _R1 == _R2 $ >
  $ _G -> all2 _R1 == all2 _R2 $;
StepHypRefExpression
1 hyp _Rh 
_G -> _R1 == _R2
2 1 eleq2d 
_G -> (x, y e. _R1 <-> x, y e. _R2)
3 2 imeq2d 
_G -> (nth n l2 = suc y -> x, y e. _R1 <-> nth n l2 = suc y -> x, y e. _R2)
4 3 imeq2d 
_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)
5 4 aleqd 
_G -> (A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. _R1) <-> A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. _R2))
6 5 aleqd 
_G -> (A. x A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. _R1) <-> A. x A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. _R2))
7 6 aleqd 
_G -> (A. n A. x A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. _R1) <-> A. n A. x A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. _R2))
8 7 aneq2d 
_G ->
  (len l1 = len l2 /\ A. n A. x A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. _R1) <->
    len l1 = len l2 /\ A. n A. x A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. _R2))
9 8 abeqd 
_G ->
  {l2 | len l1 = len l2 /\ A. n A. x A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. _R1)} ==
    {l2 | len l1 = len l2 /\ A. n A. x A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. _R2)}
10 9 sabeqd 
_G ->
  S\ l1, {l2 | len l1 = len l2 /\ A. n A. x A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. _R1)} ==
    S\ l1, {l2 | len l1 = len l2 /\ A. n A. x A. y (nth n l1 = suc x -> nth n l2 = suc y -> x, y e. _R2)}
11 10 conv all2 
_G -> all2 _R1 == all2 _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)