theorem abeq {x: nat} (p q: wff x): $ A. x (p <-> q) -> {x | p} == {x | q} $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
elab |
y e. {x | p} <-> [y / x] p |
| 2 |
|
elab |
y e. {x | q} <-> [y / x] q |
| 3 |
|
nfsb1 |
F/ x [y / x] p |
| 4 |
|
nfsb1 |
F/ x [y / x] q |
| 5 |
3, 4 |
nfbi |
F/ x [y / x] p <-> [y / x] q |
| 6 |
|
sbq |
x = y -> (p <-> [y / x] p) |
| 7 |
|
sbq |
x = y -> (q <-> [y / x] q) |
| 8 |
6, 7 |
bieqd |
x = y -> (p <-> q <-> ([y / x] p <-> [y / x] q)) |
| 9 |
5, 8 |
ealeh |
A. x (p <-> q) -> ([y / x] p <-> [y / x] q) |
| 10 |
1, 2, 9 |
bitr4g |
A. x (p <-> q) -> (y e. {x | p} <-> y e. {x | q}) |
| 11 |
10 |
iald |
A. x (p <-> q) -> A. y (y e. {x | p} <-> y e. {x | q}) |
| 12 |
11 |
conv eqs |
A. x (p <-> q) -> {x | p} == {x | q} |
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),
axs_set
(elab)