theorem rlistind {x l a: nat} (n: nat) (px: wff x) (p0 pn: wff) (pl: wff l)
(ps: wff l a):
$ x = n -> (px <-> pn) $ >
$ x = 0 -> (px <-> p0) $ >
$ x = l -> (px <-> pl) $ >
$ x = l |> a -> (px <-> ps) $ >
$ p0 $ >
$ pl -> ps $ >
$ pn $;
Step | Hyp | Ref | Expression |
1 |
|
hyp hn |
x = n -> (px <-> pn) |
2 |
|
hyp h0 |
x = 0 -> (px <-> p0) |
3 |
|
hyp hl |
x = l -> (px <-> pl) |
4 |
|
hyp hs |
x = l |> a -> (px <-> ps) |
5 |
|
hyp h1 |
p0 |
6 |
5 |
a1i |
T. -> p0 |
7 |
|
hyp h2 |
pl -> ps |
8 |
7 |
anwr |
T. /\ pl -> ps |
9 |
1, 2, 3, 4, 6, 8 |
rlistindd |
T. -> pn |
10 |
9 |
trud |
pn |
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)