Metamath Proof Explorer


Theorem zfauscl

Description: Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep , we invoke the Axiom of Extensionality (indirectly via vtocl ), which is needed for the justification of class variable notation.

If we omit the requirement that y not occur in ph , we can derive a contradiction, as notzfaus shows. (Contributed by NM, 21-Jun-1993)

Ref Expression
Hypothesis zfauscl.1
|- A e. _V
Assertion zfauscl
|- E. y A. x ( x e. y <-> ( x e. A /\ ph ) )

Proof

Step Hyp Ref Expression
1 zfauscl.1
 |-  A e. _V
2 eleq2
 |-  ( z = A -> ( x e. z <-> x e. A ) )
3 2 anbi1d
 |-  ( z = A -> ( ( x e. z /\ ph ) <-> ( x e. A /\ ph ) ) )
4 3 bibi2d
 |-  ( z = A -> ( ( x e. y <-> ( x e. z /\ ph ) ) <-> ( x e. y <-> ( x e. A /\ ph ) ) ) )
5 4 albidv
 |-  ( z = A -> ( A. x ( x e. y <-> ( x e. z /\ ph ) ) <-> A. x ( x e. y <-> ( x e. A /\ ph ) ) ) )
6 5 exbidv
 |-  ( z = A -> ( E. y A. x ( x e. y <-> ( x e. z /\ ph ) ) <-> E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) ) )
7 ax-sep
 |-  E. y A. x ( x e. y <-> ( x e. z /\ ph ) )
8 1 6 7 vtocl
 |-  E. y A. x ( x e. y <-> ( x e. A /\ ph ) )