Theorem cleljust 2164

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 2159 with the class variables in wcel 2158. (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 1536	. . 3 |- ( x e. y -> A. z x e. y )
2		elequ1 2162	. . 3 |- ( z = x -> ( z e. y <-> x e. y ) )
3	1, 2	equsex 1738	. 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 1502

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 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-13 2160

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)