theorem apprlamed2 (G: wff) (a c d: nat) {x: nat} (f b: nat x):
$ G -> c e. a $ >
$ G /\ x = c -> b = d $ >
$ G -> f = \. x e. a, b -> f @ c = d $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
appneq1 |
f = \. x e. a, b -> f @ c = (\. x e. a, b) @ c |
| 2 |
1 |
eqeq1d |
f = \. x e. a, b -> (f @ c = d <-> (\. x e. a, b) @ c = d) |
| 3 |
|
hyp e |
G /\ x = c -> b = d |
| 4 |
|
hyp h |
G -> c e. a |
| 5 |
3, 4 |
apprlamed |
G -> (\. x e. a, b) @ c = d |
| 6 |
2, 5 |
syl5ibrcom |
G -> f = \. x e. a, b -> f @ c = d |
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
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)