theorem cbvalh {x y: nat} (p q: wff x y):
$ F/ y p $ >
$ F/ x q $ >
$ x = y -> (p <-> q) $ >
$ A. x p <-> A. y q $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
hyp h1 |
F/ y p |
| 2 |
1 |
nfal |
F/ y A. x p |
| 3 |
|
hyp h2 |
F/ x q |
| 4 |
|
hyp e |
x = y -> (p <-> q) |
| 5 |
3, 4 |
ealeh |
A. x p -> q |
| 6 |
2, 5 |
ialdh |
A. x p -> A. y q |
| 7 |
3 |
nfal |
F/ x A. y q |
| 8 |
|
eqcom |
y = x -> x = y |
| 9 |
4, 8 |
syl |
y = x -> (p <-> q) |
| 10 |
9 |
bicomd |
y = x -> (q <-> p) |
| 11 |
1, 10 |
ealeh |
A. y q -> p |
| 12 |
7, 11 |
ialdh |
A. y q -> A. x p |
| 13 |
6, 12 |
ibii |
A. x p <-> A. y q |
Axiom use
axs_prop_calc
(ax_1,
ax_2,
ax_3,
ax_mp),
axs_pred_calc
(ax_gen,
ax_4,
ax_5,
ax_6,
ax_7,
ax_10,
ax_11,
ax_12)