Theorem ssab1 ≪ | index | src | ≫ 

theorem ssab1 (A: set) {x: nat} (p: wff x):
  $ A. x (p -> x e. A) <-> {x | p} C_ A $;
StepHypRefExpression
1 nfv 
F/ y p -> x e. A
2 nfab1 
FS/ x {x | p}
3 2 nfel2 
F/ x y e. {x | p}
4 nfv 
F/ x y e. A
5 3, 4 nfim 
F/ x y e. {x | p} -> y e. A
6 elab 
y e. {x | p} <-> [y / x] p
7 sbq 
x = y -> (p <-> [y / x] p)
8 6, 7 syl6bbr 
x = y -> (p <-> y e. {x | p})
9 eleq1 
x = y -> (x e. A <-> y e. A)
10 8, 9 imeqd 
x = y -> (p -> x e. A <-> y e. {x | p} -> y e. A)
11 1, 5, 10 cbvalh 
A. x (p -> x e. A) <-> A. y (y e. {x | p} -> y e. A)
12 11 conv subset 
A. x (p -> x e. A) <-> {x | p} C_ 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)