theorem all2eqd (_G: wff) (_R1 _R2: set):
$ _G -> _R1 == _R2 $ >
$ _G -> all2 _R1 == all2 _R2 $;
Step | Hyp | Ref | Expression |
1 |
|
biidd |
_G -> (len l1 = len l2 <-> len l1 = len l2) |
2 |
|
biidd |
_G -> (nth n l1 = suc x <-> nth n l1 = suc x) |
3 |
|
biidd |
_G -> (nth n l2 = suc y <-> nth n l2 = suc y) |
4 |
|
eqidd |
_G -> x, y = x, y |
5 |
|
hyp _Rh |
_G -> _R1 == _R2 |
6 |
4, 5 |
eleqd |
_G -> (x, y e. _R1 <-> x, y e. _R2) |
7 |
3, 6 |
imeqd |
_G -> (nth n l2 = suc y -> x, y e. _R1 <-> nth n l2 = suc y -> x, y e. _R2) |
8 |
2, 7 |
imeqd |
_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) |
9 |
8 |
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)) |
10 |
9 |
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)) |
11 |
10 |
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)) |
12 |
1, 11 |
aneqd |
_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)) |
13 |
12 |
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)} |
14 |
13 |
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)} |
15 |
14 |
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)