theorem dftrue2 (n: nat): $ bool n -> (true n <-> n = 1) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bi1 |
(bool n <-> true n -> n = 1) -> bool n -> true n -> n = 1 |
| 2 |
|
bool01 |
bool n <-> n = 0 \/ n = 1 |
| 3 |
2 |
conv ne, or, true |
bool n <-> true n -> n = 1 |
| 4 |
1, 3 |
ax_mp |
bool n -> true n -> n = 1 |
| 5 |
|
d1ne0 |
1 != 0 |
| 6 |
|
neeq1 |
n = 1 -> (n != 0 <-> 1 != 0) |
| 7 |
6 |
conv true |
n = 1 -> (true n <-> 1 != 0) |
| 8 |
5, 7 |
mpbiri |
n = 1 -> true n |
| 9 |
8 |
a1i |
bool n -> n = 1 -> true n |
| 10 |
4, 9 |
ibid |
bool n -> (true n <-> n = 1) |
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_peano
(peano1,
peano2,
peano5,
addeq,
add0,
addS)