theorem eqsucext (a b: nat) {x: nat}:
$ a = b <-> A. x (a = suc x <-> b = suc x) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqeq1 |
a = b -> (a = suc x <-> b = suc x) |
| 2 |
1 |
iald |
a = b -> A. x (a = suc x <-> b = suc x) |
| 3 |
|
anr |
A. x (a = suc x <-> b = suc x) /\ a = 0 -> a = 0 |
| 4 |
|
exsuc |
a != 0 <-> E. x a = suc x |
| 5 |
4 |
conv ne |
~a = 0 <-> E. x a = suc x |
| 6 |
|
exsuc |
b != 0 <-> E. x b = suc x |
| 7 |
6 |
conv ne |
~b = 0 <-> E. x b = suc x |
| 8 |
|
exeq |
A. x (a = suc x <-> b = suc x) -> (E. x a = suc x <-> E. x b = suc x) |
| 9 |
5, 7, 8 |
bitr4g |
A. x (a = suc x <-> b = suc x) -> (~a = 0 <-> ~b = 0) |
| 10 |
9 |
bi2d |
A. x (a = suc x <-> b = suc x) -> ~b = 0 -> ~a = 0 |
| 11 |
10 |
con4d |
A. x (a = suc x <-> b = suc x) -> a = 0 -> b = 0 |
| 12 |
11 |
imp |
A. x (a = suc x <-> b = suc x) /\ a = 0 -> b = 0 |
| 13 |
3, 12 |
eqtr4d |
A. x (a = suc x <-> b = suc x) /\ a = 0 -> a = b |
| 14 |
13 |
exp |
A. x (a = suc x <-> b = suc x) -> a = 0 -> a = b |
| 15 |
|
eexb |
E. x a = suc x -> a = b <-> A. x (a = suc x -> a = b) |
| 16 |
|
anr |
(a = suc x <-> b = suc x) /\ a = suc x -> a = suc x |
| 17 |
|
bi1 |
(a = suc x <-> b = suc x) -> a = suc x -> b = suc x |
| 18 |
17 |
imp |
(a = suc x <-> b = suc x) /\ a = suc x -> b = suc x |
| 19 |
16, 18 |
eqtr4d |
(a = suc x <-> b = suc x) /\ a = suc x -> a = b |
| 20 |
19 |
exp |
(a = suc x <-> b = suc x) -> a = suc x -> a = b |
| 21 |
20 |
alimi |
A. x (a = suc x <-> b = suc x) -> A. x (a = suc x -> a = b) |
| 22 |
15, 21 |
sylibr |
A. x (a = suc x <-> b = suc x) -> E. x a = suc x -> a = b |
| 23 |
5, 22 |
syl5bi |
A. x (a = suc x <-> b = suc x) -> ~a = 0 -> a = b |
| 24 |
14, 23 |
casesd |
A. x (a = suc x <-> b = suc x) -> a = b |
| 25 |
2, 24 |
ibii |
a = b <-> A. x (a = suc x <-> b = suc 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_peano
(peano1,
peano2,
peano5)