Theorem cleljust 2181

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 2176 with the class variables in wcel 2175. (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 1548	. . 3 |- ( x e. y -> A. z x e. y )
2		elequ1 2179	. . 3 |- ( z = x -> ( z e. y <-> x e. y ) )
3	1, 2	equsex 1750	. 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 1514

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-13 2177

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)