Theorem cleljust 2208

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 2203 with the class variables in wcel 2202. (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 1574	. . 3 |- ( x e. y -> A. z x e. y )
2		elequ1 2206	. . 3 |- ( z = x -> ( z e. y <-> x e. y ) )
3	1, 2	equsex 1776	. 2 |- ( E. z ( z = x /\ z e. y ) <-> x e. y )
4	3	bicomi 132	1 |- ( x e. y <-> E. z ( z = x /\ z e. y ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-13 2204

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)