Theorem sbsid ≪ | index | src | ≫ 

theorem sbsid {x: nat} (A: set x): $ S[x / x] A == A $;
StepHypRefExpression
1 bitr 
(y e. S[x / x] A <-> [x / x] y e. A) -> ([x / x] y e. A <-> y e. A) -> (y e. S[x / x] A <-> y e. A)
2 elsbs 
y e. S[x / x] A <-> [x / x] y e. A
3 1, 2 ax_mp 
([x / x] y e. A <-> y e. A) -> (y e. S[x / x] A <-> y e. A)
4 sbid 
[x / x] y e. A <-> y e. A
5 3, 4 ax_mp 
y e. S[x / x] A <-> y e. A
6 5 eqri 
S[x / x] A == 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)