theorem isfbd (F: set) (G: wff) (a b b2: nat):
$ G -> isfun F $ >
$ G -> a, b e. F $ >
$ G -> (a, b2 e. F <-> b = b2) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
hyp h1 |
G -> isfun F |
| 2 |
1 |
anwl |
G /\ a, b2 e. F -> isfun F |
| 3 |
|
hyp h2 |
G -> a, b e. F |
| 4 |
3 |
anwl |
G /\ a, b2 e. F -> a, b e. F |
| 5 |
|
anr |
G /\ a, b2 e. F -> a, b2 e. F |
| 6 |
2, 4, 5 |
isfd |
G /\ a, b2 e. F -> b = b2 |
| 7 |
|
preq2 |
b = b2 -> a, b = a, b2 |
| 8 |
7 |
anwr |
G /\ b = b2 -> a, b = a, b2 |
| 9 |
8 |
eleq1d |
G /\ b = b2 -> (a, b e. F <-> a, b2 e. F) |
| 10 |
3 |
anwl |
G /\ b = b2 -> a, b e. F |
| 11 |
9, 10 |
mpbid |
G /\ b = b2 -> a, b2 e. F |
| 12 |
6, 11 |
ibida |
G -> (a, b2 e. F <-> b = b2) |
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,
ax_8),
axs_the
(theid,
the0),
axs_peano
(peano2,
addeq,
muleq)