Theorem sbseht ≪ | index | src | ≫ 

theorem sbseht {x: nat} (a: nat) (A B: set x):
  $ FS/ x B $ >
  $ A. x (x = a -> A == B) -> S[a / x] A == B $;
StepHypRefExpression
1 elsbs 
y e. S[a / x] A <-> [a / x] y e. A
2 hyp h 
FS/ x B
3 2 nfel2 
F/ x y e. B
4 3 sbeht 
A. x (x = a -> (y e. A <-> y e. B)) -> ([a / x] y e. A <-> y e. B)
5 imim2 
(A == B -> (y e. A <-> y e. B)) -> (x = a -> A == B) -> x = a -> (y e. A <-> y e. B)
6 eleq2 
A == B -> (y e. A <-> y e. B)
7 5, 6 ax_mp 
(x = a -> A == B) -> x = a -> (y e. A <-> y e. B)
8 7 alimi 
A. x (x = a -> A == B) -> A. x (x = a -> (y e. A <-> y e. B))
9 4, 8 syl 
A. x (x = a -> A == B) -> ([a / x] y e. A <-> y e. B)
10 1, 9 syl5bb 
A. x (x = a -> A == B) -> (y e. S[a / x] A <-> y e. B)
11 10 eqrd 
A. x (x = a -> A == B) -> S[a / x] 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)