theorem rappsab {x: nat} (A: set x): $ (S\ x, A) @' x == A $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
rappsabs |
(S\ x, A) @' a1 == S[a1 / x] A |
| 2 |
|
rappeq2 |
a1 = x -> (S\ x, A) @' a1 == (S\ x, A) @' x |
| 3 |
|
eqcom |
a1 = x -> x = a1 |
| 4 |
|
sbsq |
x = a1 -> A == S[a1 / x] A |
| 5 |
4 |
eqscomd |
x = a1 -> S[a1 / x] A == A |
| 6 |
3, 5 |
rsyl |
a1 = x -> S[a1 / x] A == A |
| 7 |
2, 6 |
eqseqd |
a1 = x -> ((S\ x, A) @' a1 == S[a1 / x] A <-> (S\ x, A) @' x == A) |
| 8 |
1, 7 |
sbeth |
(S\ x, A) @' x == A |
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)