Theorem ssab2 ≪ | index | src | ≫ 

theorem ssab2 (A: set) {x: nat} (p: wff x):
  $ A. x (x e. A -> p) <-> A C_ {x | p} $;
StepHypRefExpression
1 nfv 
F/ y x e. A -> p
2 nfv 
F/ x y e. A
3 nfab1 
FS/ x {x | p}
4 3 nfel2 
F/ x y e. {x | p}
5 2, 4 nfim 
F/ x y e. A -> y e. {x | p}
6 eleq1 
x = y -> (x e. A <-> y e. A)
7 elab 
y e. {x | p} <-> [y / x] p
8 sbq 
x = y -> (p <-> [y / x] p)
9 7, 8 syl6bbr 
x = y -> (p <-> y e. {x | p})
10 6, 9 imeqd 
x = y -> (x e. A -> p <-> y e. A -> y e. {x | p})
11 1, 5, 10 cbvalh 
A. x (x e. A -> p) <-> A. y (y e. A -> y e. {x | p})
12 11 conv subset 
A. x (x e. A -> p) <-> A C_ {x | p}

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)