theorem prellame (b: nat) {x: nat} (y z: nat) (a: nat x):
$ x = y -> a = b $ >
$ y, z e. \ x, a <-> z = b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(y, z e. \ x, a <-> z = N[y / x] a) -> (z = N[y / x] a <-> z = b) -> (y, z e. \ x, a <-> z = b) |
| 2 |
|
prellams |
y, z e. \ x, a <-> z = N[y / x] a |
| 3 |
1, 2 |
ax_mp |
(z = N[y / x] a <-> z = b) -> (y, z e. \ x, a <-> z = b) |
| 4 |
|
eqeq2 |
N[y / x] a = b -> (z = N[y / x] a <-> z = b) |
| 5 |
|
hyp h |
x = y -> a = b |
| 6 |
5 |
sbne |
N[y / x] a = b |
| 7 |
4, 6 |
ax_mp |
z = N[y / x] a <-> z = b |
| 8 |
3, 7 |
ax_mp |
y, z e. \ x, a <-> z = b |
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)