theorem rappsabed (B: set) (G: wff) (a: nat) {x: nat} (A: set x):
$ G /\ x = a -> A == B $ >
$ G -> (S\ x, A) @' a == B $;
Step | Hyp | Ref | Expression |
1 |
|
rappsabs |
(S\ x, A) @' a == S[a / x] A |
2 |
|
ax_6 |
E. x x = a |
3 |
|
nfv |
F/ x G |
4 |
|
nfsbs1 |
FS/ x S[a / x] A |
5 |
|
nfsv |
FS/ x B |
6 |
4, 5 |
nfeqs |
F/ x S[a / x] A == B |
7 |
|
sbsq |
x = a -> A == S[a / x] A |
8 |
7 |
anwr |
G /\ x = a -> A == S[a / x] A |
9 |
|
hyp h |
G /\ x = a -> A == B |
10 |
8, 9 |
eqstr3d |
G /\ x = a -> S[a / x] A == B |
11 |
10 |
exp |
G -> x = a -> S[a / x] A == B |
12 |
3, 6, 11 |
eexdh |
G -> E. x x = a -> S[a / x] A == B |
13 |
2, 12 |
mpi |
G -> S[a / x] A == B |
14 |
1, 13 |
syl5eqs |
G -> (S\ x, A) @' a == 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)