theorem cbvabd {x y: nat} (G: wff) (p: wff x) (q: wff y):
$ G /\ x = y -> (p <-> q) $ >
$ G -> {x | p} == {y | q} $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
cbvabs |
{x | p} == {y | [y / x] p} |
| 2 |
1 |
a1i |
G -> {x | p} == {y | [y / x] p} |
| 3 |
|
sbet |
A. x (x = y -> (p <-> q)) -> ([y / x] p <-> q) |
| 4 |
|
hyp h |
G /\ x = y -> (p <-> q) |
| 5 |
4 |
ialda |
G -> A. x (x = y -> (p <-> q)) |
| 6 |
3, 5 |
syl |
G -> ([y / x] p <-> q) |
| 7 |
6 |
abeqd |
G -> {y | [y / x] p} == {y | q} |
| 8 |
2, 7 |
eqstrd |
G -> {x | p} == {y | 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)