Theorem cleljust 1854

Description: When the class variables of set theory are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 1434 with the class variables in wcel 1433. (Contributed by NM, 28-Jan-2004.)

Assertion

Ref	Expression
cleljust	|- ( x e. y <-> E. z ( z = x /\ z e. y ) )

Distinct variable groups:   x, z   y, z

Proof of Theorem cleljust

Step	Hyp	Ref	Expression
1		ax-17 1459	. . 3 |- ( x e. y -> A. z x e. y )
2		elequ1 1640	. . 3 |- ( z = x -> ( z e. y <-> x e. y ) )
3	1, 2	equsex 1656	. 2 |- ( E. z ( z = x /\ z e. y ) <-> x e. y )
4	3	bicomi 130	1 |- ( x e. y <-> E. z ( z = x /\ z e. y ) )

Syntax hints:   /\ wa 102   <-> wb 103   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-13 1444  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)