theorem sbq {x: nat} (a: nat) (b: wff x): $ x = a -> (b <-> [a / x] b) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
ax_12 |
x = y -> b -> A. x (x = y -> b) |
| 2 |
|
anll |
x = a /\ b /\ y = a -> x = a |
| 3 |
|
anr |
x = a /\ b /\ y = a -> y = a |
| 4 |
2, 3 |
eqtr4d |
x = a /\ b /\ y = a -> x = y |
| 5 |
|
anlr |
x = a /\ b /\ y = a -> b |
| 6 |
1, 4, 5 |
sylc |
x = a /\ b /\ y = a -> A. x (x = y -> b) |
| 7 |
6 |
ialda |
x = a /\ b -> A. y (y = a -> A. x (x = y -> b)) |
| 8 |
7 |
exp |
x = a -> b -> A. y (y = a -> A. x (x = y -> b)) |
| 9 |
|
ax_6 |
E. y y = a |
| 10 |
|
nfv |
F/ y x = a |
| 11 |
|
nfal1 |
F/ y A. y (y = a -> A. x (x = y -> b)) |
| 12 |
|
nfv |
F/ y b |
| 13 |
11, 12 |
nfim |
F/ y A. y (y = a -> A. x (x = y -> b)) -> b |
| 14 |
|
eal |
A. y (y = a -> A. x (x = y -> b)) -> y = a -> A. x (x = y -> b) |
| 15 |
|
anr |
x = a /\ y = a -> y = a |
| 16 |
15 |
imim1i |
(y = a -> A. x (x = y -> b)) -> x = a /\ y = a -> A. x (x = y -> b) |
| 17 |
16 |
com12 |
x = a /\ y = a -> (y = a -> A. x (x = y -> b)) -> A. x (x = y -> b) |
| 18 |
|
eal |
A. x (x = y -> b) -> x = y -> b |
| 19 |
18 |
com12 |
x = y -> A. x (x = y -> b) -> b |
| 20 |
|
eqtr4 |
x = a -> y = a -> x = y |
| 21 |
20 |
imp |
x = a /\ y = a -> x = y |
| 22 |
19, 21 |
syl |
x = a /\ y = a -> A. x (x = y -> b) -> b |
| 23 |
17, 22 |
syld |
x = a /\ y = a -> (y = a -> A. x (x = y -> b)) -> b |
| 24 |
14, 23 |
syl5 |
x = a /\ y = a -> A. y (y = a -> A. x (x = y -> b)) -> b |
| 25 |
24 |
exp |
x = a -> y = a -> A. y (y = a -> A. x (x = y -> b)) -> b |
| 26 |
10, 13, 25 |
eexdh |
x = a -> E. y y = a -> A. y (y = a -> A. x (x = y -> b)) -> b |
| 27 |
9, 26 |
mpi |
x = a -> A. y (y = a -> A. x (x = y -> b)) -> b |
| 28 |
8, 27 |
ibid |
x = a -> (b <-> A. y (y = a -> A. x (x = y -> b))) |
| 29 |
28 |
conv sb |
x = a -> (b <-> [a / x] b) |
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_12)