Theorem sbsco ≪ | index | src | ≫ 

theorem sbsco {x y: nat} (a: nat) (A: set x):
  $ S[a / y] S[y / x] A == S[a / x] A $;
StepHypRefExpression
1 bitr 
([a / y] z e. S[y / x] A <-> [a / y] [y / x] z e. A) -> ([a / y] [y / x] z e. A <-> [a / x] z e. A) -> ([a / y] z e. S[y / x] A <-> [a / x] z e. A)
2 elsbs 
z e. S[y / x] A <-> [y / x] z e. A
3 2 sbeq2i 
[a / y] z e. S[y / x] A <-> [a / y] [y / x] z e. A
4 1, 3 ax_mp 
([a / y] [y / x] z e. A <-> [a / x] z e. A) -> ([a / y] z e. S[y / x] A <-> [a / x] z e. A)
5 sbco 
[a / y] [y / x] z e. A <-> [a / x] z e. A
6 4, 5 ax_mp 
[a / y] z e. S[y / x] A <-> [a / x] z e. A
7 6 abeqi 
{z | [a / y] z e. S[y / x] A} == {z | [a / x] z e. A}
8 7 conv sbs 
S[a / y] S[y / x] A == S[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)