theorem dfsb3 {x: nat} (a: nat) (b: wff x): $ [a / x] b <-> E. x (x = a /\ b) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
nfex1 |
F/ x E. x (x = a /\ b) |
| 2 |
|
con2b |
(A. x ~(x = a /\ b) <-> ~b) -> (b <-> ~A. x ~(x = a /\ b)) |
| 3 |
2 |
conv ex |
(A. x ~(x = a /\ b) <-> ~b) -> (b <-> E. x (x = a /\ b)) |
| 4 |
|
bitr4 |
(A. x ~(x = a /\ b) <-> A. x (x = a -> ~b)) -> ([a / x] ~b <-> A. x (x = a -> ~b)) -> (A. x ~(x = a /\ b) <-> [a / x] ~b) |
| 5 |
|
notan2 |
~(x = a /\ b) <-> x = a -> ~b |
| 6 |
5 |
aleqi |
A. x ~(x = a /\ b) <-> A. x (x = a -> ~b) |
| 7 |
4, 6 |
ax_mp |
([a / x] ~b <-> A. x (x = a -> ~b)) -> (A. x ~(x = a /\ b) <-> [a / x] ~b) |
| 8 |
|
dfsb2 |
[a / x] ~b <-> A. x (x = a -> ~b) |
| 9 |
7, 8 |
ax_mp |
A. x ~(x = a /\ b) <-> [a / x] ~b |
| 10 |
|
sbq |
x = a -> (~b <-> [a / x] ~b) |
| 11 |
10 |
bicomd |
x = a -> ([a / x] ~b <-> ~b) |
| 12 |
9, 11 |
syl5bb |
x = a -> (A. x ~(x = a /\ b) <-> ~b) |
| 13 |
3, 12 |
syl |
x = a -> (b <-> E. x (x = a /\ b)) |
| 14 |
1, 13 |
sbeh |
[a / x] b <-> E. x (x = a /\ 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_11,
ax_12)