theorem cbvexh {x y: nat} (p q: wff x y):
$ F/ y p $ >
$ F/ x q $ >
$ x = y -> (p <-> q) $ >
$ E. x p <-> E. y q $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
noteq |
(A. x ~p <-> A. y ~q) -> (~A. x ~p <-> ~A. y ~q) |
| 2 |
1 |
conv ex |
(A. x ~p <-> A. y ~q) -> (E. x p <-> E. y q) |
| 3 |
|
hyp h1 |
F/ y p |
| 4 |
3 |
nfnot |
F/ y ~p |
| 5 |
|
hyp h2 |
F/ x q |
| 6 |
5 |
nfnot |
F/ x ~q |
| 7 |
|
hyp e |
x = y -> (p <-> q) |
| 8 |
7 |
noteqd |
x = y -> (~p <-> ~q) |
| 9 |
4, 6, 8 |
cbvalh |
A. x ~p <-> A. y ~q |
| 10 |
2, 9 |
ax_mp |
E. x p <-> E. 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)