theorem sbeht {x: nat} (a: nat) (b c: wff x):
$ F/ x c $ >
$ A. x (x = a -> (b <-> c)) -> ([a / x] b <-> c) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
nfal1 |
F/ x A. x (x = a -> (b <-> c)) |
| 2 |
|
nfsb1 |
F/ x [a / x] b |
| 3 |
|
hyp h |
F/ x c |
| 4 |
2, 3 |
nfbi |
F/ x [a / x] b <-> c |
| 5 |
1, 4 |
nfim |
F/ x A. x (x = a -> (b <-> c)) -> ([a / x] b <-> c) |
| 6 |
|
sbq |
x = a -> (b <-> [a / x] b) |
| 7 |
6 |
anwl |
x = a /\ A. x (x = a -> (b <-> c)) -> (b <-> [a / x] b) |
| 8 |
|
eal |
A. x (x = a -> (b <-> c)) -> x = a -> (b <-> c) |
| 9 |
8 |
impcom |
x = a /\ A. x (x = a -> (b <-> c)) -> (b <-> c) |
| 10 |
7, 9 |
bitr3d |
x = a /\ A. x (x = a -> (b <-> c)) -> ([a / x] b <-> c) |
| 11 |
10 |
exp |
x = a -> A. x (x = a -> (b <-> c)) -> ([a / x] b <-> c) |
| 12 |
5, 11 |
eexh |
E. x x = a -> A. x (x = a -> (b <-> c)) -> ([a / x] b <-> c) |
| 13 |
|
ax_6 |
E. x x = a |
| 14 |
12, 13 |
ax_mp |
A. x (x = a -> (b <-> c)) -> ([a / x] b <-> c) |
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)