theorem sabeq {x: nat} (A B: set x): $ A. x A == B -> S\ x, A == S\ x, B $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
nfsbs1 |
FS/ x S[fst a1 / x] A |
| 2 |
|
nfsbs1 |
FS/ x S[fst a1 / x] B |
| 3 |
1, 2 |
nfeqs |
F/ x S[fst a1 / x] A == S[fst a1 / x] B |
| 4 |
|
sbsq |
x = fst a1 -> A == S[fst a1 / x] A |
| 5 |
|
sbsq |
x = fst a1 -> B == S[fst a1 / x] B |
| 6 |
4, 5 |
eqseqd |
x = fst a1 -> (A == B <-> S[fst a1 / x] A == S[fst a1 / x] B) |
| 7 |
3, 6 |
ealeh |
A. x A == B -> S[fst a1 / x] A == S[fst a1 / x] B |
| 8 |
7 |
eleq2d |
A. x A == B -> (snd a1 e. S[fst a1 / x] A <-> snd a1 e. S[fst a1 / x] B) |
| 9 |
8 |
abeqd |
A. x A == B -> {a1 | snd a1 e. S[fst a1 / x] A} == {a1 | snd a1 e. S[fst a1 / x] B} |
| 10 |
9 |
conv sab |
A. x A == B -> S\ x, A == S\ x, 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)