Theorem eqval ≪ | index | src | ≫ 

theorem eqval (A: set) {x: nat}: $ A == _V <-> A. x x e. A $;
StepHypRefExpression
1 bitr3 
(_V C_ A <-> A == _V) -> (_V C_ A <-> A. x x e. A) -> (A == _V <-> A. x x e. A)
2 ssv1 
_V C_ A <-> A == _V
3 1, 2 ax_mp 
(_V C_ A <-> A. x x e. A) -> (A == _V <-> A. x x e. A)
4 biim1 
x e. _V -> (x e. _V -> x e. A <-> x e. A)
5 elv 
x e. _V
6 4, 5 ax_mp 
x e. _V -> x e. A <-> x e. A
7 6 aleqi 
A. x (x e. _V -> x e. A) <-> A. x x e. A
8 7 conv subset 
_V C_ A <-> A. x x e. A
9 3, 8 ax_mp 
A == _V <-> A. x x e. 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)